Preprint (2014)

MORSE THEORY AND LESCOP’S EQUIVARIANT

PROPAGATOR FOR 3-MANIFOLDS WITH b1 = 1 FIBERED

OVER S1

TADAYUKI WATANABE

Abstract. For a 3-manifold M with b1(M) = 1 fibered over S1 and thefiberwise gradient ξ of a fiberwise Morse function on M , we introduce thenotion of Z-path in M . A Z-path is a piecewise smooth path in M consistingof edges each of which is either a path in a critical locus of ξ or a flow line

of −ξ. Counting closed Z-paths with signs gives the Lefschetz zeta functionof M . The “moduli space” of Z-paths in M has a description like Chen’siterated integrals and gives explicitly Lescop’s equivariant propagator, whosePoincare dual generates the second cohomology of the configuration space oftwo points with rational function coefficients and which can be used to expressZ-equivariant version of Chern–Simons perturbation theory for M by countinggraphs. Counting Z-paths that connects two components in a nullhomologouslink in M gives the equivariant linking number.

1. Introduction

Chern–Simons perturbation theory for 3-manifolds was developed independently

by Axelrod–Singer ([AS]) and by Kontsevich ([Ko]). It is defined by integrations

over suitably compactified configuration spaces C2n,∞(M) of a closed 3-manifold

M and gives a strong invariant Z(M) of M whose universal formula is a formal

series of Feynman diagrams (e.g. [KT, Les1]). In the definition of Z, propagator

plays an important role. Here, a propagator is a certain closed 2-form on C2,∞(M),

which corresponds to an edge in a Feynman diagram (see [AS, Ko] for the defini-

tion of propagator, and [Les1] for a detailed exposition). The Poincare–Lefschetz

dual to a propagator is given by a relative 4-cycle in (C2,∞(M), ∂C2,∞(M)). In

a dual perspective, given three parallels P1, P2, P3 of such a 4-cycle, the algebraic

triple intersection number #P1 ∩ P2 ∩ P3 in the 6-manifold C2,∞(M) gives rise to

the 2-loop part of Z, which corresponds to the Θ-shaped Feynman diagram. For

general 3-valent graphs with 2n vertices, the intersections of codimension 2 cycles

in C2n,∞(M) give rise to invariants of M .

Propagator may not exist depending on the topology of M . For a propagator

with Q coefficients to exist, M must be a Q homology 3-sphere. If b1(M) >

0, one must improve the method to find universal perturbative invariant for M

whose value is a formal series of Feynman diagrams, which is not classical. After

Date: September 2, 2015.2000 Mathematics Subject Classification. 57M27, 57R57, 58D29, 58E05.

1

2 TADAYUKI WATANABE

Ohtsuki’s pioneering work that refines the LMO invariant significantly ([Oh1, Oh2]),

Lescop gave a topological construction of an invariant of M with b1(M) = 1 for

the 2-loop graph using configuration spaces and using similar argument as given

in Marche’s work on equivariant Casson knot invariant ([Ma]). More precisely,

she defined in [Les2] a topological invariant of M by using the equivariant triple

intersection of “equivariant propagators” in the “equivariant configuration space”

C2,Z(M) of M∗. The equivariant configuration space C2,Z(M) is an infinite cyclic

covering of the compactified configuration space C2(M). An equivariant propagator

is defined as a relative 4-cycle in (C2,Z(M), ∂C2,Z(M)) with coefficients in Q(t)

satisfying a certain boundary condition, which is described by a rational function

including the Alexander polynomial of M ([Les2, Theorem 4.8] or Theorem 1.2

below). The existence of an equivariant propagator as in [Les2] is a key to carry

out equivariant perturbation theory for 3-manifolds with b1 > 0. Indeed, she proved

that some equivalence class of the equivariant triple intersection of three equivariant

propagators gives rise to an invariant of M .

In this paper, we introduce the notion of Z-path in a 3-manifoldM with b1(M) =

1 fibered over S1 (Definition 1.4) and we construct an equivariant propagator ex-

plicitly as the chain given by the moduli space of Z-paths in M (Theorem 1.5).

Note that the existence of an equivariant propagator satisfying an explicit bound-

ary condition is proved in [Les2], whereas globally explicit cycle is not referred to

except for the case M = S2 × S1. In proving the main Theorem 1.5, we show that

the counts of closed Z-paths in M give the Lefschetz zeta function of the fibration

M (Proposition 4.10). In a sense, our construction gives a geometric derivation of

the formula for Lescop’s boundary condition. Moreover, by Theorem 1.5, we get

an explicit path-counting formula of the equivariant linking number of two compo-

nent nullhomologous link in M counting Z-paths whose endpoints are on the link

components (Theorem 4.12).

Z-path is in a sense a piecewise smooth approximation of integral curve of a

nonsingular vector field on M (see Figure 5). Let ξ be the gradient along the

fibers of a fiberwise Morse function (§1.5) of M . Roughly speaking, a Z-path in M

is a piecewise smooth path in M that is an alternating concatenation of vertical

segments and horizontal segments, where a vertical segment is a part of a flow line

of −ξ and a horizontal segment is a path in a critical locus of ξ both descending.

The explicit propagator given in this paper is useful to make some core argu-

ments in equivariant perturbation theory into explicit path-counting ones. Inspired

by the ideas of [Fu, Wa1] for construction of graph-counting invariants for homol-

ogy 3-spheres, we obtain a candidate for equivariant version of the Chern–Simons

perturbation theory for 3-manifoldsM with b1(M) = 1 fibered over S1, by counting

graphs in M each of whose edges is a Z-path for a fiberwise gradient. By explicitly

counting graphs, we obtain a surgery formula as in [Les2] of the invariant and of

the 3-manifold invariant in [Les2] for a special kind of surgery. We will write about

it in [Wa2]. We expect that our construction can be extended to 3-manifolds with

arbitrary first Betti numbers and to generic closed 1-forms, generic in the sense of

[Hu], by using a method similar to that of Pajitnov in [Pa1, Pa2].

∗In [Les2], the equivariant configuration space is denoted by C2(M).

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 3

1.1. Conventions. In this paper, manifolds and maps between them are assumed

to be smooth. By an n-dimensional chain in a manifold X , we mean a finite linear

combination of smooth maps from oriented compact n-manifolds with corners to

X . We understand a chain as a chain of smooth simplices by taking triangulations

of manifolds. We denote by ∆X the diagonal in X ×X . We follow [BT, Appendix]

for the conventions for manifolds with corners and fiber products of manifolds with

corners. Some definitions needed are summarized in Appendix A.

We represent an orientation o(X) of a manifold X by a non-vanishing section

of∧dimX

T ∗X . We will often identify∧•

T ∗xX with

∧•TxX by a (locally defined)

framing that is compatible with the orientation and treat∧•

T ∗xX like

∧•TxX . We

consider a coorientation o∗(V ) of a submanifold V of a manifold X as an orientation

of the normal bundle of V and represent it by a differential form in Γ∞(∧•

T ∗X |V ).We identify the normal bundle NV with the orthogonal complement TV ⊥ in TX ,

by taking a Riemannian metric on X . We fix orientation or coorientation of V so

that the identity

o(V ) ∧ o∗(V ) ∼ o(X)

holds, where we say that two orientations o and o′ are equivalent (o ∼ o′) if they are

related by multiple of a positive function. o(V ) determines o∗(V ) up to equivalence

and vice versa. We orient boundaries of an oriented manifold by the outward normal

first convention.

We will often write unions⋃

s∈S Vs for continuous parameters s ∈ S, such as

real numbers, as

∫

s∈S

Vs. When the parameter is at most countable, we will write

the unions as∑

s∈S

Vs or Vs1 + Vs2 + · · · .

1.2. Lefschetz zeta function. We shall recall a few definitions and notations

before stating the main result. Let Σ be a closed manifold. For a diffeomorphism

ϕ : Σ→ Σ, its Lefschetz zeta function ζϕ(t) is defined by the formula

ζϕ(t) = exp

(∞∑

k=1

L(ϕk)

ktk

)∈ Q[[t]],

where L(ϕk) is the Lefschetz number of the iteration ϕk, or the count of fixed points

of ϕk counted with appropriate signs if ϕ is generic. The following product formula

is a consequence of the Lefschetz trace formula.

(1.2.1) ζϕ(t) =dimΣ∏

i=0

det(1− tϕ∗i)(−1)i+1

,

where ϕ∗i : Hi(Σ;Q)→ Hi(Σ;Q) is the induced map from ϕ. See e.g. [Pa2, 9.2.1].

In this paper, we will often consider the logarithmic derivative of ζϕ(t):

(1.2.2)d

dtlog ζϕ(t) =

ζ′ϕ(t)

ζϕ(t)=

dimΣ∑

i=0

(−1)iTr ϕ∗i

1− tϕ∗i.

4 TADAYUKI WATANABE

1.3. Equivariant configuration spaces. We recall some definitions from [Les2].

LetM be a closed oriented Riemannian 3-manifold with b1(M) = 1, let κ :M → S1

be a map that induces an isomorphism H1(M)/Torsion → H1(S1) and let M be

the standard infinite cyclic covering that is connected. Let π : M → M be the

covering projection. Let κ : M → R be the lift of κ and let t : M → M be

the diffeomorphism that generate the group of covering transformations and that

satisfies for every x ∈ M ,

κ(tx) = κ(x) + 1.

Let M ×Z M be the quotient of M × M by the equivalence relation that identifies

x× y with tx× ty. We denote the equivalence class of x× y by x×Z y. The natural

map π : M ×Z M → M ×M is an infinite cyclic covering. By abuse of notation,

we denote by t the generator of the group of covering transformations of M ×Z M

that acts as follows.

t(x ×Z y) = (t−1x)×Z y = x×Z (ty).

The compactified configuration space C2(M) is the compactification of M ×M \ ∆M that is obtained from M × M by blowing-up the diagonal ∆M . See

Appendix B for the definition of blow-up. Roughly, the blow-up replaces ∆M with

its normal sphere bundle. The boundary ∂C2(M) is canonically identified with the

unit tangent bundle ST (M) of M . More precisely, let N∆Mbe the total space of

the normal bundle of ∆M in M ×M . We fix a framing τ : TM → R3×M , which is

compatible with the orientation of M . The framing of M induces an isomorphism

(1.3.1) φ : N∆M→ R3 ×∆M

of oriented vector bundles. Namely, if e1, e2, e3 is the basis of TxM induced by τ

and if e1, e2, e3, e′1, e′2, e′3 is the induced basis of TxM ⊕ TxM , then (T(x,x)∆M )⊥

is spanned by e′1 − e1, e′2 − e2, e′3 − e3 and φ is defined by

φ(a1(e′1 − e1) + a2(e

′2 − e2) + a3(e

′3 − e3), (x, x)) = (a1, a2, a3)× (x, x).

Then φ induces a diffeomorphism Bℓ0(N∆M) → Bℓ0(R

3) ×∆M , under which the

boundary of Bℓ0(N∆M) corresponds to ∂Bℓ0(R

3) × ∆M = S2 × ∆M ≈ S2 ×M .

We denote φ−1(S2 ×∆M ) by ST (M). Note that the blowing-up does not depend

on the choice of τ .

Let ∆M = π−1(∆M ). The equivariant configuration space C2,Z(M) is defined by

C2,Z(M) = Bℓ∆M(M ×Z M),

the blow-up of M ×Z M along ∆M . The boundary of C2,Z(M) is canonically

identified with Z× ST (M) =∐

i∈Z tiST (M).

1.4. Lescop’s equivariant propagator. Let K be an oriented knot in M such

that 〈[dκ], [K]〉 = 1. Let Λ = Q[t, t−1] and let Q(t) be the field of fractions of Λ.

Then H∗(C2,Z(M)) is naturally a graded Λ-module.

Theorem 1.1 (Lescop [Les2, Proposition 2.12]). For any i ∈ Z,

Hi(C2,Z(M))⊗Λ Q(t) ∼= Hi−2(M ;Q)⊗Q Q(t).

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 5

H3(C2,Z(M))⊗Λ Q(t) = Q(t)[ST (K)],

H2(C2,Z(M))⊗Λ Q(t) = Q(t)[ST (∗)],where ST (K) is the restriction of the S2-bundle ST (M) on K.

Consider the exact sequence

H4(C2,Z(M), ∂C2,Z(M))⊗ΛQ(t)∂→ H3(∂C2,Z(M))⊗ΛQ(t)

i∗→ H3(C2,Z(M))⊗ΛQ(t),

where i∗ is the map induced by the inclusion†.

Theorem 1.2 (Lescop [Les2, Theorem 4.8]). Let τ : TM → R3 ×M be a trivial-

ization of TM and let sτ :M → ST (M) be a section induced by τ that sends M to

v×M for a fixed v ∈ S2. Suppose that sτ |K agrees with the unit tangent vectors

of K. Then‡

(1.4.1) i∗[sτ (M)] = −(1 + t

1− t +t∆′(M)

∆(M)

)i∗[ST (K)]

in H3(C2,Z(M)) ⊗Λ Q(t), where ∆(M) is the Alexander polynomial of M nor-

malized so that ∆(M)(1) = 1 and ∆(M)(t−1) = ∆(M)(t). Hence, there exists a

4-dimensional Q(t)-chain Q such that

∂Q = sτ (M) +

(1 + t

1− t +t∆′(M)

∆(M)

)ST (K).

Lescop calls such a Q(t)-chain Q an equivariant propagator. The equivari-

ant intersection pairing with Q detects all classes in H2(C2,Z(M)) ⊗Λ Q(t) =

Q(t)[ST (∗)]. More generally, we will call a 4-dimensional relative Q(t)-cycle Q in

(C2,Z(M), ∂C2,Z(M)) such that the boundary condition is satisfied inH3(∂C2,Z(M))⊗Λ

Q(t) an equivariant propagator.

1.5. Fiberwise Morse function. Let M be a closed oriented Riemannian 3-

manifold with b1(M) = 1 fibered over S1 and let κ : M → S1 be the projec-

tion of the fibration. Suppose that the fiber of κ is path-connected and oriented.

A fiberwise Morse function is a smooth function f : M → R whose restriction

fs = f |κ−1(s) : κ−1(s) → R is Morse for all s ∈ S1. A generalized Morse function

(GMF) is a smooth function on a manifold with only Morse or birth-death singu-

larities ([Ig1, Appendix]). A fiberwise GMF is a smooth function f :M → R whose

restriction fs : κ−1(s)→ R is a GMF for all s ∈ S1. The fiberwise gradient ξ for a

fiberwise GMF f is the vector field on M such that the restriction ξ(s) = ξ|κ−1(s)

agrees with gradfs for each s.

A critical locus of a fiberwise GMF is a component in the subset ofM consisting

of all the critical points of fs, s ∈ S1. The family of sections dfs : Tκ−1(s) →Rs∈S1 defines a smooth section of the fiberwise cotangent bundle (Ker dκ)∗. Then

†Since Q(t) is a torsion-free Λ-module, one has the isomorphism Hi(C∗(X) ⊗Λ Q(t)) ∼=

Hi(X) ⊗Λ Q(t) of Λ-modules for any Z-space X, by the universal coefficient theorem.‡The sign in the formula (1.4.1) seems different from that of [Les2]. This is because the

homological action t of the knot in [Les2] is our t−1. Note that

1 + t−1

1− t−1+

t−1∆′(M)(t−1)

∆(M)(t−1)= −

(1 + t

1− t+

t∆′(M)

∆(M)

)

6 TADAYUKI WATANABE

Figure 1. (1) Cerf’s graphic for S1-family of smooth functions

and (2), (3) birth-death cancellation.

the union of critical loci is identified with the intersection of dfss∈S1 with the

zero section of (Ker dκ)∗, which is generically a 1-dimensional submanifold of M .

A critical locus is decomposed into finitely many intervals by birth-death points.

We also call such a segment a critical locus, abusing the notation. For a critical

locus as a segment or as a closed curve without birth-death points, we consider its

index as the Morse index of the intersection point of the locus with a generic fiber,

which is a Morse critical point.

We will need an oriented fiberwise GMF, where a fiberwise GMF is said to be

oriented if the bundles of negative eigenspaces of the Hessians along the fibers over

all the critical loci are oriented and if each birth-death pair (p, q) near a birth-

death locus has incidence number 1 (or Mpq consists of one point and positively

cooriented (see §3.3 for the meaning of this formula)).

For a critical locus p = p(s)s∈S1 of the fiberwise gradient ξ of a fiberwise Morse

function f , we denote by Dp = Dp(ξ) and Ap = Ap(ξ) the descending manifold

loci and the ascending manifold loci respectively. Namely, for each s ∈ S1, let

Dp(s)(ξ(s)) and Ap(s)(ξ(s)) (ξ(s) = ξ|κ−1(s)) be the descending and the ascending

manifolds of the critical point p(s) of fs. Then we define Dp(ξ) =⋃

s Dp(s)(ξ(s))

and Ap(ξ) =⋃

s Ap(s)(ξ(s)), which are generically submanifolds ofM . For a smooth

function σ : S1 → R, the level surface locus (for σ) is the subset L =⋃

s∈S1 L(s),

L(s) = f−1s (σ(s)) ⊂ κ−1(s).

If a pair of different critical loci p, q is such that ind p = ind q = 1 and if ξ is

generic, then Dp and Aq may intersect transversally at finitely many values of κ.

The intersection of Dp and Aq is then a flow line along ξ between p and q. Such an

intersection is called a 1/1-intersection ([HW]). It is known that a 1/1-intersection

corresponds to a 1-handle slide (e.g. [Mi, Theorem 7.6]).

Proposition 1.3. There exists an oriented fiberwise Morse function f : M → R

for the fibration κ :M → S1.

Proof. According to Cerf [Ce, Ch I.3] or the Framed function theorem of K. Igusa

[Ig1, Theorem 1.6] (see also [Ig2, Theorem 4.6.3]), there exists an oriented fiberwise

GMF f : M → R. The graph of critical values of fs forms a diagram in R × S1

(Cerf’s graphic). See Figure 1 (1) for an example. In a graphic, Morse critical loci

correspond to arcs and birth-death singularities correspond to beaks.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 7

Figure 2

Figure 3

If there is a pair of beaks as in Figure 1 (2), we may apply the Birth-death

cancellation lemma [Ig1, Proposition A.2.3] of K. Igusa to eliminate the pair of

beaks by deforming f within the space of smooth functions on M to a fiberwise

GMF with less birth-death points. Namely, let J = (c, d) ⊂ S1 be a small interval

such that a pair (v1, v2) of birth-death points as in Figure 1 (2) is included in

κ−1(J). According to the Birth-death cancellation lemma, the pair (v1, v2) can be

cancelled if there exists a smooth section σ : [κ(v1), κ(v2)]→ κ−1[κ(v1), κ(v2)] of κ

such that σ(κ(v1)) = v1 and σ(κ(v2)) = v2. From the assumption that the fiber of

κ is path-connected and by the obstruction theory, the result follows.

Here, for orientability of the descending manifolds of the result, one may need

to introduce a 1-parameter family as in Figure 2 that reverses the orientation of

a descending manifold. Such a 1-parameter family can be introduced in a small

neighborhood of a critical locus of index 2 without changing the topology of M

and the pair of beaks cancels with the new pair of beaks, as in Figure 3. (See [Ig1,

p.436] for detail.)

Finally, we must check that any beaks in a graphic can be arranged to form pairs

of beaks as in Figure 1(2). By the Beak lemma of Cerf ([Ce, Ch. IV, §3], see also

[La, Theorem 1.3]), this can be achieved. Hence all beaks can be eliminated and

the result is as desired.

1.6. Z-paths. We fix an oriented fiberwise Morse function f :M → R onM and its

gradient ξ along the fibers that satisfies the parametrized Morse–Smale condition,

i.e., the descending manifold loci and the ascending manifold loci are mutually

transversal in M . Let f : M → R denote the Z-invarint lift f = f π and let ξ

denote the lift of ξ. We say that a piecewise smooth embedding σ : [µ, ν] → M

is vertical if Imσ is included in a single fiber of κ and say that σ is horizontal if

Imσ is included in a critical locus of f . We say that a vertical embedding (resp.

horizontal embedding) σ : [µ, ν] → M is descending if f(σ(µ)) ≥ f(σ(ν)) (resp.

κ(σ(µ)) ≤ κ(σ(ν))).A flow-line of −ξ is a piecewise smooth embedding σ : [µ, ν]→ M such that for

each T ∈ [µ, ν] that is not in the preimage of the union of critical loci, dσT (∂∂T

) is

a multiple of (−ξ)σ(T ) by a positive real number.

8 TADAYUKI WATANABE

Figure 4. (1) Cerf’s graphic enriched by positions of 1/1-

intersections. (2) A Z-path.

Definition 1.4. Let x, y be two points of M such that κ(x) ≥ κ(y). A Z-path from

x to y is a sequence γ = (σ1, σ2, . . . , σn), n ≥ 1, where

(1) For each i, σi is either vertical or horizontal.

(2) For each i, σi is a descending embedding [µi, νi]→ M for some real numbers

µi, νi.

(3) If σi is vertical, then σi is a flow line of −ξ. If σi is horizontal, then µi < νi.

(4) σ1(µ1) = x, σn(νn) = y.

(5) σi(νi) = σi+1(µi+1) for 1 ≤ i < n.

(6) If σi is vertical (resp. horizontal) and if i < n, then σi+1 is horizontal (resp.

vertical).

(7) If n = 1, then µ1 < ν1.§.

We say that two Z-paths are equivalent if they differ only by reparametrizations on

segments. A Z-path in M is defined as the composition of a Z-path in M with the

covering projection π. (See Figure 4 for an example of a Z-path.)

1.7. Main result. Let M Z2 (ξ) be the set of equivalence classes of all Z-paths in

M . It will turn out that there is a natural structure of non-compact manifold on

M Z2 (ξ). The Z-action γ 7→ tnγ on a path induces a free Z-action on M Z

2 (ξ). Let

M Z2 (ξ)Z be the quotient of M Z

2 (ξ) by the Z-action. For the fiberwise gradient ξ of

f , let ξ be the nonsingular vector field −ξ+gradκ on M . Let sξ :M → ST (M) be

the normalization ξ/‖ξ‖ of the section ξ. Now we state the main theorem of this

paper, which gives an explicit equivariant propagator.

Theorem 1.5 (Theorem 4.7, Corollary 4.11). Let M be the mapping torus of an

orientation preserving diffeomorphism ϕ : Σ → Σ of closed, connected, oriented

surface Σ.

(1) There is a natural closure MZ

2 (ξ)Z of M Z2 (ξ)Z that has the structure of a

countable union of smooth compact manifolds with corners.

(2) Suppose that κ induces an isomorphism H1(M)/Torsion ∼= H1(S1). Let

b : MZ

2 (ξ)Z → M×Z M be the evaluation map, which assigns the endpoints.

Let Bℓb−1(∆M )(M

Z

2 (ξ)Z) denote the blow-up of MZ

2 (ξ)Z along b−1(∆M ).

Then b induces a map Bℓb−1(∆M)(M

Z

2 (ξ)Z)→ C2,Z(M) and it represents a

§Z-path appears very similar to “flow line with cascades”, defined before in [Fr] in relation to

Morse-Bott theory. We think that the purpose and the origin for the two notions are different

(see also §1.8). Yet it might be interesting to consider a unified generalization of both notions.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 9

4-dimensional Q(t)-chain Q(ξ) in C2,Z(M) that satisfies the identity

[∂Q(ξ)] = [sξ(M)] +tζ′ϕζϕ

[ST (K)]

in H3(∂C2,Z(M))⊗ΛQ(t). Thus Q(ξ) is an equivariant propagator. (A pre-

cise formula for ∂Q(ξ) at the chain level is given in Theorem 4.7.) More-

over, for a product C(t) of cyclotomic polynomials, C(t)∆(M)Q(ξ) is a

Λ-chain.

By [Les2, Proposition 4.5] and by the identity ∆(M) = c t−g(Σ) det(1− tϕ∗1) for

fibered 3-manifold (c = det(1 − ϕ∗1)−1. Note that b1(M) = 1 iff det(1 − ϕ∗1) 6= 0

(e.g., [Ri])), the formula in (2) recovers the formula for ∂Q in Theorem 1.2 in

H3(∂C2,Z(M)) ⊗Λ Q(t). This together with the formula in (2) shows that the

Lefschetz zeta function ζϕ can be considered as the obstruction to extending sξ(M)

to a relative Q(t)-cycle in C2,Z(M).

Note that it is not straightforward that the substitution of our chain Q(ξ) into

Lescop’s formula for the equivariant triple intersection gives rise to a topological

invariant of M because in [Les2], the proof of invariance is essentially based on the

assumption that the boundary of equivariant propagator concentrates on sτ (M)

and on tiST (K) for a given knot in M . Since we consider the union of critical loci

of ξ instead of a knot, the identity in Theorem 1.5 holds only in homology. So a

modification of the proof is required to get an invariant by using Q(ξ). See [Wa2]

for detail.

1.8. Motivation for the definition of Z-path. We give an informal illustration

of how Z-paths arise naturally, with a simple example. Let M = S2 × S1 and

consider M as the mapping torus of a generic diffeomorphism ϕ : S2 → S2 isotopic

to the identity. More precisely, suppose that ϕ is the gradient flow Φs−gradh for

a Morse function h : S2 → R and for a constant s > 0. Let M be the quotient

space of S2 × [0, 1] obtained by identifying (x, 0) with (ϕ(x), 1) for all x ∈ S2. The

rightward unit vector field ∂∂s

on S2× [0, 1] induces an rightward nonsingular vector

field ξ on M . Then ξ is of the form −α gradh+ ξ0 for a constant α > 0, where ξ0is the gradient for the projection S2×S1 → S1 with respect to the product metric.

Since h has finitely many critical points, ξ has finitely many simple closed orbits.

If we consider analogous to [Fu], a candidate for the propagator would be the

moduli space

M2(ξ) = (x, y) ∈M ×M ; ∃T > 0, y = ΦT

ξ(x).

This is a non-compact 4-dimensional manifold immersed in M ×M in a very com-

plicated way. It is hard to deal with M2(ξ) in the following sense. To define Z-

equivariant version of Chern–Simons perturbation theory, we would like to consider,

for example, the triple intersection of the moduli spaces M2(ξ1),M2(ξ2),M2(ξ3) in

M ×M for a generic triple (ξ1, ξ2, ξ3) of rightward vector fields, which corresponds

to the Θ-graph. However, since M2(ξ) is non-compact, it is unclear that the triple

intersection number is well-defined, even if counted for a fixed homotopy class of

10 TADAYUKI WATANABE

Figure 5. An integral curve is approximated by a Z-path. The

horizontal straight lines are the lifts of closed orbits of ξ of index

2 and 1.

Θ-graphs in M . For example, nullhomotopic Θ-graph may wind around arbitrarily

many times in M .

Here, Lescop’s work developed in [Les2] (given after Marche’s work on equivari-

ant Casson knot invariant [Ma]) is crucial. She considered the equivariant triple

intersection of three Q(t)-chains as in Theorem 1.2 and observed that it works

nicely for the purpose of constructing an invariant. That the triple intersection is

well-defined is then obvious from definition. To utilize Lescop’s work, we considered

deforming M2(ξ) into a Q(t)-chain.

Now let us return to the example given above. The lift γ of an integral curve γ of

ξ in M is as in Figure 5 (1). Now deform ξ = −α gradh+ξ0 to ξρ = −α gradh+ρ ξ0,

where ρ : M → R is a smooth bump function such that ρ = 1 on each closed orbit

of ξ and supported on a small neighborhood N of the union of all closed orbits of

ξ. After the deformation, ξρ is vertical outside N and an integral curve of ξρ will

become as in Figure 5 (2). If the support of ρ gets very small, an integral curve of

ξρ can be approximated by a Z-path, as in Figure 5 (3).

1.9. Organization. The rest of this paper is organized as follows. In §2, we definethe moduli space M

Z

2 (ξ)Z of Z-paths in M and study its piecewise smooth structure.

In §3, we fix (co)orientation of MZ

2 (ξ)Z. In §4, we make the moduli space MZ

2 (ξ)Zinto a Q(t)-chain Q(ξ) in the equivariant configuration space. The explicit formula

for the boundary of Q(ξ) is given in Theorem 4.7. Then the main theorem follows as

a corollary of Theorem 4.7. In Appendix A, some definitions on smooth manifolds

with corners are recalled. In Appendix B, the definition of blow-up is recalled.

2. Moduli space of Z-paths

In this section, we define the moduli space MZ

2 (ξ)Z of Z-paths in M (Defi-

nition 2.7) and study its piecewise smooth structure (Lemma 2.5). An element

γ = (σ1, . . . , σn) in M Z2 (ξ) may be described by the following data.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 11

(1) An increasing sequence s1 < · · · < sn of the parameters of vertical segments

and/or of initial/final endpoints.

(2) The positions of the initial point x of σ1 and the final point y of σn.

(3) A sequence z1, . . . , zr of points on the vertical segments σi1 , . . . , σir that

connect two critical loci.

Thus, M Z2 (ξ) may be locally described as a subset of the product of spaces of

(vertical) flow-lines of −ξ.

2.1. Moduli space of vertical flow-lines. Let M , f : M → R, f : M → R and

ξ be as in §1.6. We define

M2(ξ) = (x, y) ∈ M × M ; κ(x) = κ(y), y = ΦT

−ξ(x) for some T > 0.

We shall construct a natural closure of M2(ξ).

In a graphic of a fiberwise Morse function on M , there are intersection points

between curves of critical values. We may assume that the intersections of the curves

are all in general positions. We call an intersection of two curves in a graphic a level

exchange bifurcation. We assume that f and ξ are generic so that level exchange

bifurcations and 1/1-intersections occur at different parameters in S1.

Suppose that u1, u2, . . . , ur ∈ S1 = R/Z are the parameters at which the level

exchange bifurcations of f occur, cyclically ordered as u1 < u2 < · · · < ur < u1in S1. Choose a small number ε > 0 and let I2j−1 = (uj − 2ε, uj + 2ε) and

I2j = (uj + ε, uj+1− ε), where ur+1 = u1+1 = u1 and we identify an open interval

of length < 1 with its image in S1. We assume without loss of generality that ε is

small so that there are no 1/1-intersections in I2j−1 for all j. Then S1 is covered

by finitely many open intervals:

S1 =

2r⋃

j=1

Ij .

If there are no level exchange bifurcations of f , we consider S1 as I1 ∪ I2/∼, whereI1 = ∅ and I2 = (−ε, 1 + ε). Moreover, we consider the lifts

Ij+2rk = Ij + k (1 ≤ j ≤ 2r, k ∈ Z)

in R. Then Ij ; j ∈ Z covers R. By considering the lifts MIj+2rk= κ−1(Ij+2rk) of

MIj , the Z-covering M of M can be covered by pieces MIjj∈Z.

2.1.1. Cutting MIj into pieces. For the open cover Ij ; j ∈ Z of R given above, we

write Ij = (aj , bj). Let p1, p2, . . . , pN, pi = pi(s)s∈Ij , be the set of all critical

loci of f |MIjnumbered so that f(p1(aj)) < f(p2(aj)) < · · · < f(pN (aj)). We fix a

sufficiently small number η > 0 (see below for how small η has to be). We define

the submanifolds Γ(j)i , W

(j)i and L

(j)i of MIj as follows.

If j is even, then we define smooth functions γi : Ij → R by γi(s) = fs(pi(s))− ηfor 1 ≤ i ≤ N and by γN+1(s) = fs(pN (s)) + η for i = N +1. We define Γ

(j)i , W

(j)i

and L(j)i by

Γ(j)i =

⋃

s∈Ij

f−1s (γi+1(s)), W

(j)i =

⋃

s∈Ij

f−1s [γi(s), γi+1(s)], L

(j)i =

⋃

s∈Ij

f−1s (γi(s)).

12 TADAYUKI WATANABE

Figure 6. A cover of M from a graphic.

We will often omit the superscript (j) for simplicity.

If j is odd, then suppose that pk and pk+1 are the critical loci such that fs(pk(uj)) =

fs(pk+1(uj)) and such that faj(pk(aj)) < faj

(pk+1(aj)) and fbj (pk(bj)) > fbj (pk+1(bj)).

We refer to this k as kj . For 1 ≤ i < kj , we define γi : Ij → R as γi(s) =

fs(pi(s)) − η. For kj < i ≤ N , we define γi(s) = fs(pi+1(s)) − η. For i = kj , we

define γkj: Ij → R as a smooth function such that

(1) fs(pkj(s)) > γkj

(s) and fs(pkj+1(s)) > γkj(s) for all s ∈ Ij ,

(2) γkj(s) > fs(pkj−1(s)) for all s ∈ Ij .

(3) γkj(aj) = faj

(pkj(aj))− η, γkj

(bj) = fbj (pkj+1(bj))− η.For i = N + 1, we choose γN+1 so that γN+1(s) ≥ maxifs(pi(s)) + η. We define

L(j)i =

⋃s∈Ij

f−1s (γi(s)), and

Γ(j)i =

⋃

s∈Ij

f−1s (γi+1(s)), W

(j)i =

⋃

s∈Ij

f−1s [γi(s), γi+1(s)]

Here, we choose η so that the following conditions are satisfied.

(1) η is less than fs(pk+1(s))−fs(pk(s))2 for all k such that 1 ≤ k ≤ N − 1 and for

all s ∈ I2j .(2) η is less than fs(pk+1(s))−fs(pk(s))

2 for all k such that 1 ≤ k ≤ N − 1 except

k2j−1 and for all s ∈ I2j−1

(3) η is less thanfs(pk2j−1+2(s))−fs(pk2j−1

(s))

2 for all s ∈ I2j−1.

2.1.2. The definition of M 2(ξ). Let ξj = ξ|MIj. For a pair of subsets A,B of MIj ,

we put M2(ξj ;A,B) = M2(ξj) ∩ (A×B). Then we have

M2(ξj) =⋃

0≤ℓ≤k≤N

M2(ξj ;Wk,Wℓ).

For 1 ≤ ℓ ≤ k ≤ N , there is a natural embedding

ψkℓ : M2(ξj ;Wk,Wℓ)→Wk × Lk × Lk−1 × · · · × Lℓ+1 ×Wℓ,

defined by ψkℓ(x, y) = (x, zk, zk−1, . . . , zℓ+1, y), where zi ∈ Li is the unique inter-

section point of the flow line between x and y with Li. Then M2(ξj) is canonically

diffeomorphic to the union of the images ψkℓ(M2(ξj ;Wk,Wℓ)) (1 ≤ ℓ ≤ k ≤ N)

glued together by the diffeomorphisms

ψkℓ ψ−1k+1,ℓ : ψk+1,ℓ(M2(ξj ;Lk+1,Wℓ))→ ψkℓ(M2(ξj ;Lk+1,Wℓ)),

ψkℓ ψ−1k,ℓ−1 : ψk,ℓ−1(M2(ξj ;Wk, Lℓ))→ ψkℓ(M2(ξj ;Wk, Lℓ)).

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 13

Note that ψkℓψ−1k+1,ℓ and ψkℓψ−1

k,ℓ−1 agree with the maps induced from the projec-

tions πkℓ(x, zk+1, zk, . . . , zℓ+1, y) = (x, zk, . . . , zℓ+1, y), ρkℓ(x, zk, . . . , zℓ+1, zℓ, y) =

(x, zk, . . . , zℓ+1, y).

Let

(2.1.1) M 2(ξj ;Wk,Wℓ) = ψkℓ(M2(ξj ;Wk,Wℓ)),

where the closure is taken in the codomain of ψkℓ. The projections πkℓ and ρkℓinduce diffeomorphisms

ψk+1,ℓ(M2(ξj ;Lk+1,Wℓ))→ ψkℓ(M2(ξj ;Lk+1,Wℓ)),

ψk,ℓ−1(M2(ξj ;Wk, Lℓ))→ ψkℓ(M2(ξj ;Wk, Lℓ)).(2.1.2)

Definition 2.1. We define

(2.1.3) M 2(ξj) =⋃

1≤ℓ≤k≤N

M 2(ξj ;Wk,Wℓ),

where the pieces are glued together by the diffeomorphisms of (2.1.2), and we define

M 2(ξ) =⋃

j∈Z

M 2(ξj),

where M 2(ξj−1) and M 2(ξj) are glued together by identifying sequences of points

corresponding to flow lines of ξj or of ξj−1 in κ−1(Ij−1 ∩ Ij).Let

b : M 2(ξ)→ M × Mbe the continuous map that gives the pair of endpoints of a flow line (possibly

broken, see below). For subsets A of Wk and B of Wℓ, let

(2.1.4) M 2(ξj ;A,B) = ψkℓ(M2(ξj ;A,B)) ⊂ A× Lk × · · · × Lℓ+1 ×B.This is consistent with (2.1.1). Note that this may depend on the choices of k and

ℓ when A ⊂ Lk or B ⊂ Lℓ+1, but it becomes well-defined if it is considered as a

subspace of M 2(ξj).

For a pair (x, y) of distinct points of MIj − Σ(ξj) such that κ(x) = κ(y), a

(r times) broken flow line between x and y is a sequence γ0, γ1, . . . , γr (r ≥ 1) of

integral curves of −ξj satisfying the following conditions:

(1) The domain of γ0 is [0,∞), the domain of γr is (−∞, 0] and the domain of

γi, 1 ≤ i ≤ r − 1, is R.

(2) γ0(0) = x, γr(0) = y.

(3) There is a sequence q1, q2, . . . , qr of distinct critical loci of ξj such that

limT→−∞ γi(T ) ∈ qi and limT→∞ γi−1(T ) ∈ qi (1 ≤ i ≤ r).A broken flow line (γ0, γ1, . . . , γr) between x and y is determined by the boundary

points x, y and intersection points of γi with level surfaces that lie between qi and

qi+1. More precisely, a broken flow line between x ∈ Wk and y ∈ Wℓ is uniquely

determined by a point of Wk ×Lk × · · · ×Lℓ+1 ×Wℓ up to reparametrizations and

conversely a broken flow line between x ∈ Wk and y ∈ Wℓ determines a point of

Wk ×Lk × · · · ×Lℓ+1×Wℓ. So we may identify a broken flow line between x ∈Wk

and y ∈Wℓ with a point of Wk ×Lk × · · · ×Lℓ+1×Wℓ and call the latter a broken

flow sequence.

14 TADAYUKI WATANABE

For generic ξ, the intersection Dp(ξ) ∩ Aq(ξ) is transversal in M and hence is

a manifold. There is a free R-action on Dp(ξ) ∩ Aq(ξ) defined by x 7→ ΦT

−ξ(x)

(T ∈ R). Put

Mpq = Mpq(ξ) = (Dp(ξ) ∩Aq(ξ))/R.

This space is locally represented as the intersection of Dp(ξ) ∩ Aq(ξ) with a level

surface of f . The dimension of the manifold Mpq is ind p− ind q.

Proposition 2.2. M 2(ξ) satisfies the following conditions.

(1) M 2(ξ)− b−1(∆M) is a manifold with corners.

(2) b induces a diffeomorphism IntM 2(ξ) → M2(ξ), where Int denotes the

codimension 0 stratum.

(3) The codimension r stratum of M 2(ξ)− b−1(∆M) consists of r times broken

flow sequences. The codimension r stratum of M 2(ξ)− b−1(∆M) for r ≥ 1

is canonically diffeomorphic to

∫

s∈R

∑

q1∈Σ(ξ)

Aq1(ξ(s))×Dq1(ξ(s))−∑

q1∈Σ(ξ)

∆q1 (if r = 1)

∫

s∈R

∑

q1,...,qr∈Σ(ξ)q1,...,qr distinct

Aq1(ξ(s))×Mq1q2(ξ(s))× · · · ×Mqr−1qr (ξ(s))×Dqr (ξ(s)) (if r ≥ 2)

The proof of Proposition 2.2 will be given in §2.3.Let p1, . . . , pN denote the set of all critical loci of f numbered so that f(p1(0)) <

f(p2(0)) < · · · < f(pN (0)). The expression of the codimension r stratum of

M 2(ξ)− b−1(∆M) in Proposition 2.2 can be abbreviated as∫

s∈R

X(s)× Ω(s)× · · · × Ω(s)︸ ︷︷ ︸r−1

×tY (s), where

X(s) = (Ap1(ξ) Ap2(ξ) · · · ApN(ξ))(s),

Y (s) = (Dp1(ξ) Dp2(ξ) · · · DpN(ξ))(s),

Ω(s) =

∅ Mp1p2(ξ) Mp1p3(ξ) · · · Mp1pN(ξ)

Mp2p1(ξ) ∅ Mp2p3(ξ) · · · Mp2pN(ξ)

Mp3p1(ξ) Mp3p2(ξ) ∅ Mp3pN(ξ)

......

. . ....

MpNp1(ξ) MpNp2(ξ) MpNp3(ξ) · · · ∅

(s),

and the direct product of matrices is given by matrix multiplication with the mul-

tiplications and the sums given respectively by direct products and disjoint unions.

The codimension r stratum of a matrix of stratified spaces is the matrix of codi-

mension r strata.

The following two propositions are the restrictions of Proposition 2.2.

Proposition 2.3. Let p be a critical locus of ξ and let Dp(ξ) = b−1(p × M),

A p(ξ) = b−1(M × p).(1) Dp(ξ) (resp. A p(ξ)) is a manifold with corners.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 15

(2) b induces a diffeomorphism IntDp(ξ)→ Dp(ξ) (resp. IntA p(ξ)→ Ap(ξ)).

(3) The codimension r stratum of tY = (Dp1(ξ) Dp2(ξ) · · · DpN(ξ)) (resp.

X = (A p1(ξ) A p2(ξ) · · · A pN(ξ))) for r ≥ 1 is canonically diffeomorphic

to∫

s∈R

Ω(s)× · · · × Ω(s)︸ ︷︷ ︸r

×tY (s) (resp.

∫

s∈R

X(s)× Ω(s)× · · · × Ω(s)︸ ︷︷ ︸r

).

Proposition 2.4. Let p, q be critical loci of ξ and let M pq(ξ) = b−1(p× q).(1) M pq(ξ) is a manifold with corners.

(2) There is a natural diffeomorphism IntM pq(ξ)→Mpq(ξ).

(3) The codimension r stratum of Ω = ((1− δij)M pipj(ξ)) for r ≥ 1 is canon-

ically diffeomorphic to∫

s∈R

Ω(s)× · · · × Ω(s)︸ ︷︷ ︸r+1

.

2.2. Moduli space of Z-paths. A fiberwise space over a space B is a space E

together with a continuous map φ : E → B. The fiber over a point s ∈ B is

E(s) = φ−1(s) ([CJ]). For two fiberwise spaces E1 = (E1, φ1) and E2 = (E2, φ2)

over B, the fiberwise product E1 ×B E2 is the subspace of E1 × E2 given by

E1 ×B E2 =

∫

s∈B

E1(s)× E2(s),

or in other words, E1 ×B E2 is the pullback of the diagram E1φ1→ B

φ2← E2.

For a sequence of fiberwise spaces over R: Ai = (Ai, φi), φi : Ai → R (i =

1, 2, . . . , n), we define the iterated integrals as∫

R

A1A2 · · ·An =

∫

s1<s2<···<sn

A1(s1)×A2(s2)× · · · ×An(sn)

= (φ1 × · · · × φn)−1(s1 < · · · < sn),∫

R

A1A2 · · ·An =

∫

s1≤s2≤···≤sn

A1(s1)×A2(s2)× · · · ×An(sn)

= (φ1 × · · · × φn)−1(s1 ≤ · · · ≤ sn).For a matrix P = (Aij) of fiberwise spaces over R, the fiber over s ∈ R is P (s) =

(Aij(s)). With this notation, the iterated integrals for matrices of fiberwise spaces

over R are defined by the same formula as above.

Let X,Y,Ω be the matrices of fiberwise spaces over R defined by

X = (Ap1(ξ) Ap2(ξ) · · · ApN(ξ)), Y = (Dp1(ξ) Dp2(ξ) · · · DpN

(ξ)),

Ω = ((1 − δij)Mpipj(ξ))1≤i,j≤N .

Then the space of Z-paths in M can be written as

MZ2 (ξ) = M2(ξ) +

∫

R

X tY +

∫

R

XΩ tY +

∫

R

XΩΩ tY + · · · ,

where the sum is disjoint union. We consider a natural closure of this space.

16 TADAYUKI WATANABE

Lemma 2.5. The space

∫

R

X Ω · · ·Ω︸ ︷︷ ︸n

tY is a disjoint union of finitely many mani-

folds with corners such that the closure of the codimension 1 stratum is given by∫

R

(∂X)Ω · · ·Ω︸ ︷︷ ︸n

tY +

n∑

i=1

∫

R

X Ω · · ·Ω︸ ︷︷ ︸i−1

(∂Ω)Ω · · ·Ω︸ ︷︷ ︸n−i

tY +

∫

R

X Ω · · ·Ω︸ ︷︷ ︸n

(∂tY )

+

∫

R

(X ×R Ω)Ω · · ·Ω︸ ︷︷ ︸n−1

tY +

n−1∑

i=1

∫

R

X Ω · · ·Ω︸ ︷︷ ︸i−1

(Ω×R Ω)Ω · · ·Ω︸ ︷︷ ︸n−i−1

tY +

∫

R

X Ω · · ·Ω︸ ︷︷ ︸n−1

(Ω×RtY ).

Proof. Let λ : X × Ω× · · · × Ω︸ ︷︷ ︸n

×tY → Rn+2 be the smooth map that is de-

fined for γ0 ∈ A q0(ξ)(s0), γ1 ∈ M q0q1(ξ)(s1), . . . , γn ∈ M qn−1qn(ξ)(sn), γn+1 ∈Dqn(ξ)(sn+1) by

λ(γ0, γ1, . . . , γn+1) = (s0, s1, . . . , sn+1).

Let σ = (s0, s1, . . . , sn+1) ∈ Rn+2 ; s0 ≤ s1 ≤ · · · ≤ sn+1. Then∫

R

X Ω · · ·Ω︸ ︷︷ ︸n

tY =

λ−1(σ). Since no pair of 1/1-intersections occur at the same fiber, every possible

perturbations of a point of ∂σ in the inward directions in σ, bounded by a small

number δ > 0, have lifts in λ−1(σ) from any point of λ−1(∂σ). Hence λ is strata

transversal to σ. This shows that λ−1(σ) is a manifold with corners whose codi-

mension r stratum is the intersection of the codimension r1 stratum of the domain

of λ and the preimage of the codimension r − r1 stratum of σ. When r = 1, we

have (r1, r − r1) = (1, 0), (0, 1). The former (resp. the latter) gives the first line

(resp. the second line) of the formula of the statement.

For n ≥ 0, we denote by Sn (resp. Tn) the first line (resp. the second line) of the

formula of the closure of the codimension 1 stratum in Lemma 2.5. The following

lemma is a corollary to Propositions 2.3 and 2.4.

Lemma 2.6. There are natural strata preserving diffeomorphisms

∂X ∼= X ×R Ω, ∂ tY ∼= Ω×RtY ,

∂Ω ∼= Ω×R Ω, ∂M 2(ξ) ∼= ∆M

+X ×RtY ,

which induce, for n ≥ 0, strata preserving diffeomorphism Sn∼= Tn+1.

We denote by S−1 the part of ∂M 2(ξ) that is identified with X ×RtY in

Lemma 2.6.

Definition 2.7. We define

MZ

2 (ξ) =

[M 2(ξ) +

∫

R

X tY +

∫

R

X Ω tY +

∫

R

X ΩΩ tY + · · ·]/∼,

where we identify Sn−1 with Tn for all n ≥ 0 by the diffeomorphisms Sn∼= Tn+1 of

Lemma 2.6. The group Z acts on MZ

2 (ξ) by the diagonal action (x1, x2, . . . , xn) 7→(tx1, tx2, . . . , txn), which is strata-preserving, and we define

MZ

2 (ξ)Z = MZ

2 (ξ)/Z.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 17

2.3. Smooth structure of the moduli space of vertical flow-lines. From now

on, we shall give a proof of the following lemma in order to prove Proposition 2.2.

Lemma 2.8. Proposition 2.2 with ξ replaced by ξj holds.

The proof is an analogue of [BH, Theorem 1] and [We].

2.3.1. Moduli space of short vertical flow-lines in a fiber. Let h : Rd → R be the

standard quadratic form h(x1, . . . , xd) = −x21

2 − · · · −x2i

2 +x2i+1

2 + · · · + x2d

2 . First,

we describe the following standard model and its closure.

M2(h) = (x, y) ∈ Rd × Rd ; y = ΦT−gradh(x) for some T ∈ (0,∞)

Lemma 2.9. M2(h) = (ρu, v)× (u, ρv); u ∈ Ri, v ∈ Rd−i, ρ ∈ (0, 1). Hence its

closure M 2(h) in Rd × Rd is

M 2(h) = (ρu, v)× (u, ρv); u ∈ Ri, v ∈ Rd−i, ρ ∈ [0, 1]and M 2(h)− 0× 0 is a manifold with boundary, whose boundary is

(0 × Rd−i)× (Ri × 0) ∪0×0 ∆Rd = (A0(h)×D0(h)) ∪0×0 ∆Rd .

Proof. Let X = (ρu, v) × (u, ρv); u ∈ Ri, v ∈ Rd−i, ρ ∈ (0, 1). Suppose that

(ρu, v)× (u, ρv) ∈X . The solution for the differential equation

γ(T ) = (−gradh)γ(T )

is γ(T ) = (γ1(0)eT , . . . , γi(0)e

T , γi+1(0)e−T , . . . , γd(0)e

−T ). If γ(0) = (ρu, v), then

γ(T ) = (ρueT , ve−T ). The system of equations ρueT = u, ve−T = ρv has a unique

solution T ≥ 0 provided that (u, v) 6= (0, 0), in which case (ρu, v)×(u, ρv) ∈M2(h).

If (u, v) = (0, 0), then (ρu, v) × (u, ρv) = (0, 0)× (0, 0) and obviously this belongs

M2(h). Conversely, for (u0, v0) × (u0eT , v0e

−T ) ∈ M2(h) (T ≥ 0), put u = u0eT

and v = v0. Then (u0, v0) × (u0eT , v0e

−T ) = (ue−T , v) × (u, ve−T ) ∈ X . This

completes the proof of X = M2(h).

For the latter assertion, consider the smooth map ϕ : [0, 1]×Ri×Rd−i → Rd×Rd

defined by ϕ(ρ, u, v) = (ρu, v)× (u, ρv). Its Jacobian matrix is

(2.3.1) Jϕ(ρ,u,v) =

u ρI O

0 O I

0 I O

v O ρI

whose rank is d + 1 unless (u, v) = (0, 0). Namely, ϕ is an immersion outside

[0, 1]× 0 × 0. Note that ϕ([0, 1]× 0 × 0) = 0 × 0. Moreover, it is easy to check

that M 2(h)−0× 0 is a submanifold with boundary. The boundary corresponds

to the image from ρ = 0, 1.

Lemma 2.10. Let 1 ≤ k ≤ N and suppose that j is even and that s ∈ Ij . Then

(i) M 2(ξj ;Wk(s),Wk(s))−∆Wk(s) is a submanifold of Wk(s)×Wk(s) with cor-

ners, with

∂(M 2(ξj ;Wk(s),Wk(s))−∆Wk(s))

=((Apk

(s) ∩Wk(s))× (Dpk(s) ∩Wk(s))

)− (pk(s), pk(s))

+ M2(ξj ;Wk(s), Lk(s)) + M2(ξj ; Γk(s),Wk(s)).

18 TADAYUKI WATANABE

(ii) M 2(ξj ;Wk(s), Lk(s)) is a submanifold of Wk(s)× Lk(s) with corners, with

∂M 2(ξj ;Wk(s), Lk(s))

=((Apk

(s) ∩Wk(s))× (Dpk(s) ∩ Lk(s))

)+ M2(ξj ; Γk(s), Lk(s)) + ∆Lk(s).

(iii) M 2(ξj ; Γk(s), Lk(s)) is a submanifold of Γk(s)× Lk(s) with corners, with

∂M 2(ξj ; Γk(s), Lk(s)) = (Apk(s) ∩ Γk(s))× (Dpk

(s) ∩ Lk(s))

(iv) M 2(ξj ; Γk(s),Wk(s)) is a submanifold of Γk(s)×Wk(s) with corners, with

∂M 2(ξj ; Γk(s),Wk(s))

=((Apk

(s) ∩ Γk(s))× (Dpk(s) ∩Wk(s))

)+ M2(ξj ; Γk(s), Lk(s)) + ∆Γk(s).

Proof. Here we only prove (i). The other cases are the restrictions of this case. The

part M2(ξj ;Wk(s), Lk(s))+M2(ξj ; Γk(s),Wk(s)) is the boundary of M2(ξj ;Wk(s),Wk(s)).

To study the other ends, we define Uk ⊂Mpk(s) by the condition

−ε < −x21 − · · · − x2i + x2i+1 + · · ·+ x2d < ε

(−x21 − · · · − x2i )(x2i+1 + · · ·+ x2d) < ε

for a sufficiently small number ε > 0, where d = 2 and i is the index of pk. The clo-

sure of M2(ξj ;Uk, Uk) in Uk×Uk is the restriction of M 2(h) in Lemma 2.9. Let Uk ⊂Wk(s) be the open subset defined as the union of all the images of integral curves of

ξj that intersect Uk. By extention by the flow of ξj , one may see that the boundary

of the closure of M2(ξj ; Uk, Uk) in Uk× Uk consists of degenerate flow-lines coming

from those in ∂M 2(h) and flow-lines of ∂(Uk × Uk). Then M 2(ξj ;Wk(s),Wk(s))

is the union of M 2(ξj ; Uk, Uk) and M2(ξj ;Wk(s) −Kk(s),Wk(s) −Kk(s)), where

Kk(s) = Apk(s) ∪Dpk

(s). This completes the proof.

2.3.2. Moduli space of short vertical flow-lines in 1-parameter family.

Corollary 2.11. Let 1 ≤ k ≤ N and suppose that j is even. Then

(i) M 2(ξj ;Wk,Wk)−∆Wkis a submanifold of Wk ×Wk with corners, with

∂(M 2(ξj ;Wk,Wk)−∆Wk)

=((Apk

∩Wk)×Ij (Dpk∩Wk)

)−∆pk

+ M2(ξj ;Wk, Lk) + M2(ξj ; Γk,Wk).

(ii) M 2(ξj ;Wk, Lk) is a submanifold of Wk × Lk with corners, with

∂M 2(ξj ;Wk, Lk) =((Apk

∩Wk)×Ij (Dpk∩ Lk)

)+ M2(ξj ; Γk, Lk) + ∆Lk

.

(iii) M 2(ξj ; Γk, Lk) is a submanifold of Γk × Lk with corners, with

∂M 2(ξj ; Γk, Lk) =(Apk∩ Γk)×Ij (Dpk

∩ Lk)

(iv) M 2(ξj ; Γk,Wk) is a submanifold of Γk ×Wk with corners, with

∂M 2(ξj ; Γk,Wk) =((Apk

∩ Γk)×Ij (Dpk∩Wk)

)+ M2(ξj ; Γk, Lk) + ∆Γk

.

Proof. Since there are no bifurcation for Apkand Dpk

in Wk, there are fiber-

preserving diffeomorphisms M 2(ξj ;Wk,Wk) ∼= M 2(ξj ;Wk(s),Wk(s)) × Ij ,M 2(ξj ;Wk, Lk) ∼= M 2(ξj ;Wk(s), Lk(s))×Ij , M 2(ξj ; Γk, Lk) ∼= M 2(ξj ; Γk(s), Lk(s))×

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 19

Ij , M 2(ξj ; Γk,Wk) ∼= M 2(ξj ; Γk(s),Wk(s)) × Ij of manifolds with corners. These

together with Lemma 2.10 above completes the proof.

When j is odd and i 6= kj , Corollary 2.11 hold without any change.

Lemma 2.12. Suppose that j is odd and that i = kj. Then the same assertions as

Corollary 2.11 hold.

Proof. Since the descending manifold loci and the ascending manifold loci between

the two different critical loci inWi are disjoint, the set of flow-lines that are close to

one critical locus is disjoint from that of flow-lines that are close to another critical

locus. So the smooth structures on the corners can be studied separately for each

critical locus and the result follows.

Next, we shall prove the following lemma.

Lemma 2.13. M 2(ξj ;Wk,Wℓ) (1 ≤ ℓ < k ≤ N , definition in (2.1.1)) is a sub-

manifold of Wk ×Lk × Lk−1 × · · · ×Lℓ+1 ×Wℓ with corners, whose codimension r

stratum for r ≥ 1 consists of r− s times broken flow sequences σ with s ≤ 2 events

in the following list happening.

• The initial endpoint of σ lies in ∂Wk.

• The terminal endpoint of σ lies in ∂Wℓ.

To prove Lemma 2.13, we shall prove the following lemma.

Lemma 2.14. For k− ℓ− 1 ≥ 0, the moduli space M 2(ξj ;Wk, Lℓ+1) is a subman-

ifold of Wk × Lk × Lk−1 × · · · × Lℓ+1 with corners, whose codimension r stratum

for r ≥ 1 consists of r − s times broken flow sequences σ with the following event

happening s ≤ 1 times: The initial endpoint of σ lies in ∂Wk.

We prove Lemma 2.14 by induction on k− ℓ− 1. For k− ℓ− 1 = 0, Lemma 2.14

has been proved in Lemma 2.10. Let us consider the case k − ℓ − 1 = 1, i.e.

M 2(ξj ;Wk, Lk−1). Let i2 : M2(ξj ;Wk, Lk) → Lk and i1 : M2(ξj ;Lk, Lk−1) →Lk denote the maps induced from projections pr2 : Wk × Lk → Lk and pr1 :

Lk × Lk−1 → Lk respectively. By these maps, we consider M2(ξj ;Wk, Lk) and

M2(ξj ;Lk, Lk−1) as fiberwise spaces overLk. Then the moduli spaceM2(ξj ;Wk, Lk−1)

is identified with the fiberwise product M2(ξj ;Wk, Lk)×LkM2(ξj ;Lk, Lk−1). It is

easy to see that i2 and i1 are transversal and hence by Proposition A.2 the fiber

product is a manifold with boundary.

Lemma 2.15. The smooth extensions i2 : M 2(ξj ;Wk, Lk)→ Lk and i1 : M 2(ξj ;Lk, Lk−1)→Lk of the projections i2 and i1 respectively are strata transversal. Hence the fiber

product

M 2(ξj ;Wk, Lk)×LkM 2(ξj ;Lk, Lk−1) ⊂Wk × Lk × Lk × Lk−1

is a manifold with corners, whose strata are as follows.

(0) The codimension 0 stratum is M2(ξj ; IntWk, Lk)×LkM2(ξj ;Lk, Lk−1).

(1) The codimension 1 stratum is the union of ∂1M 2(ξj ;Wk, Lk)×LkM2(ξj ;Lk, Lk−1)

and M2(ξj ; IntWk, Lk)×Lk∂1M 2(ξj ;Lk, Lk−1), where ∂r denotes the codi-

mension r stratum.

20 TADAYUKI WATANABE

(2) The codimension 2 stratum is ∂1M 2(ξj ;Wk, Lk)×Lk∂1M 2(ξj ;Lk, Lk−1).

Proof. For simplicity, we assume that j is even. If either zk ∈ i2(M2(ξj ;Wk, Lk))

or zk ∈ i1(M2(ξj ;Lk, Lk−1)), then zk is a regular value of one of i2 and i1.

Indeed, if for example zk = i2(x, zk), (x, zk) ∈ i2(M2(ξj ;Wk, Lk)), then there

is a small open neighborhood O of zk in Lk such that TzkO and the tangent

space of the gradient line at x parametrizes T(x,zk)M2(ξj ;Wk, Lk). Obviously,

di2 : T(x,zk)M2(ξj ;Wk, Lk) → TzkLk = TzkO is surjective. This shows that i2and i1 are transversal between a codimension 0 stratum and any strata.

If zk ∈ i2(∂M 2(ξj ;Wk, Lk)−∂M2(ξj ;Wk, Lk)) and zk ∈ i1(∂M 2(ξj ;Lk, Lk−1)),

then we have i−12 (zk) = (Apk

∩Wk)×zk and i−11 (zk) = zk×(Dpk−1

∩Lk−1). The

images of the orthogonal complement of T(x,zk)i−12 (zk) in T(x,zk)∂M 2(ξj ;Wk, Lk)

and of T(zk,y)i−11 (zk) in T(zk,y)∂M 2(ξj ;Lk, Lk−1) under the differentials di2 and di1

agree with Tzk(Dpk∩ Lk) and Tzk(Apk−1

∩ Lk) respectively. By the parametrized

Morse–Smale condition for ξj , these images span TzkLk. This shows that i2 and i1are transversal between codimension 1 strata. Now the lemma follows by applying

Proposition A.2.

The following lemma proves Lemma 2.14 for k − ℓ− 1 = 1.

Lemma 2.16. We have

(2.3.2) M 2(ξj ;Wk, Lk−1) = pr(M 2(ξj ;Wk, Lk)×Lk

M 2(ξj ;Lk, Lk−1)),

where pr : M 2(ξj ;Wk, Lk)×LkM 2(ξj ;Lk, Lk−1)→Wk ×Lk ×Lk−1 is the embed-

ding (x, zk, zk, zk−1) 7→ (x, zk, zk−1). Hence M 2(ξj ;Wk, Lk−1) is a manifold with

corners, whose strata are as follows.

(0) The codimension 0 stratum is pr(M2(ξj ; IntWk, Lk)×Lk

M2(ξj ;Lk, Lk−1)).

(1) The codimension 1 stratum is the union of

pr(∂1M 2(ξj ;Wk, Lk)×Lk

M2(ξj ;Lk, Lk−1))and

pr(M2(ξj ; IntWk, Lk)×Lk

∂1M 2(ξj ;Lk, Lk−1)).

(2) The codimension 2 stratum is pr(∂1M 2(ξj ;Wk, Lk)×Lk

∂1M 2(ξj ;Lk, Lk−1)).

Proof. First, pr takes M 2(ξj ;Wk, Lk)×LkM 2(ξj ;Lk, Lk−1) diffeomorphically onto

its image because the map pr is smooth and there is a smooth section γ : Wk ×Lk × Lk−1 →Wk ×∆Lk

× Lk−1 ⊂Wk × Lk × Lk × Lk−1 of pr.

Since M 2(ξj ;Wk, Lk−1) is the closure of ψk,k−1(M2(ξj ;Wk, Lk−1)) inWk×Lk×Lk−1 and since γ gives a homeomorphismWk×Lk×Lk−1 ≈Wk×∆Lℓ

×Lk−1, it suf-

fices to show that the closure of γ(ψk,k−1(M2(ξj ;Wk, Lk−1))) = M2(ξj ;Wk, Lk)×Lk

M2(ξj ;Lk, Lk−1) agrees with M 2(ξj ;Wk, Lk)×LkM 2(ξj ;Lk, Lk−1) to see (2.3.2).

This follows from Proposition A.4 because the codimension 0 stratum of latter space

agrees with the former one.

Proof of Lemma 2.14. We assume that Lemma 2.14 holds true for k− ℓ− 1 = p ≤N−3 and we show that Lemma 2.14 holds true for k−ℓ−1 = p+1. By assumption,

the moduli space M 2(ξj ;Wk, Lℓ+1) is a manifold with corners, whose strata are as

described in Lemma 2.14. Then by exactly the same argument as in Lemmas 2.15

and 2.16, one may see the following, which completes the proof.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 21

(1) By Proposition A.2, the fiber product M 2(ξj ;Wk, Lℓ+1)×Lℓ+1M 2(ξj ;Lℓ+1, Lℓ) ⊂

(Wk ×Lk ×Lk−1 × · · · × Lℓ+1)× (Lℓ+1 ×Lℓ) has the structure of a manifold with

corners, whose codimension r stratum is the union of ∂rM 2(ξj ;Wk, Lℓ+1) ×Lℓ+1

M2(ξj ;Lℓ+1, Lℓ) and ∂r−1M 2(ξj ;Wk, Lℓ+1)×Lℓ+1∂1M 2(ξj ;Lℓ+1, Lℓ).

(2) By Proposition A.4,

M 2(ξj ;Wk, Lℓ) = pr(M 2(ξj ;Wk, Lℓ+1)×Lℓ+1

M 2(ξj ;Lℓ+1, Lℓ))

where pr : (Wk×Lk×Lk−1×· · ·×Lℓ+1)×(Lℓ+1×Lℓ)→Wk×Lk×Lk−1×· · ·×Lℓ+1×Lℓ is the projection (x, zk, zk−1, . . . , zℓ+1, zℓ+1, zℓ) 7→ (x, zk, zk−1, . . . , zℓ+1, zℓ), which

embeds M 2(ξj ;Wk, Lℓ+1)×Lℓ+1M 2(ξj ;Lℓ+1, Lℓ).

Proof of Lemma 2.13. By replacing M 2(ξj ;Lℓ+1, Lℓ) in the proof of Lemma 2.14

with M 2(ξj ;Lℓ+1,Wℓ), one may see by Proposition A.4 that M 2(ξj ;Wk,Wℓ) agrees

with the projection of the fiber product M 2(ξj ;Wk, Lℓ+1)×Lℓ+1M 2(ξj ;Lℓ+1,Wℓ)

which is a manifold with corners as desired.

Proof of Lemma 2.8. Now we know from Lemma 2.13 that M 2(ξj) is the union of

moduli spaces M 2(ξj ;Wk,Wℓ) (1 ≤ ℓ ≤ k ≤ N) that are manifolds with corners,

glued together by diffeomorphisms (2.1.2). The result is a manifold with corners

(see Lemma 2.10 for the reason of exclusion of the diagonal). This proves the

property (1). The property (2) is immediate from the definition of M 2(ξj).

Since the diffeomorphisms (2.1.2) are strata preserving (Appendix A) in both

directions, no new corners will appear under the gluing. The diffeomorphisms

induce gluings between strata of the same codimensions and of the same type. For

example, the component of r times broken flow sequences in M 2(ξj ,Wk,Wℓ) is

glued together along M 2(ξj ;Lk+1,Wℓ) with the component of r times broken flow

sequences in M 2(ξj ;Wk+1,Wℓ). This proves the property (3).

Proof of Proposition 2.2. There is a canonical strata-preserving diffeomorphism be-

tween the restrictions of MZ

2 (ξj) and MZ

2 (ξj+1) on Ij ∩ Ij+1. According to Defini-

tion 2.1, MZ

2 (ξ) is obtained by gluing the pieces MZ

2 (ξj) by such diffeomorphisms.

No new corners appear under the gluing.

3. Convention for (co)orientations

3.1. (Co)orientations of descending and ascending manifolds. Let p be a

critical locus of ξ, Dp be the descending manifold locus of p and Ap be the ascending

manifold locus of p. Let x be a point of p and let Σ(s) be the fiber of κ over

s = κ(x). Let o(M) denote a volume form on M giving the orientation and we

write o(M) = π∗o(M).

We define the orientation of Σ(s) at x by o(Σ(s))x = ι(grad κ) o(M)x.

Let Dp(s) = Dp ∩Σ(s). For an arbitrarily given orientation o(Dp(s))x of Dp(s),

we define the orientation of the ascending manifold Ap(s) = Ap ∩Σ(s) by the rule

o(Dp(s))x ∧ o(Ap(s))x = o(Σ(s))x.

22 TADAYUKI WATANABE

We define the orientations of Dp and Ap by

o(Dp)x = dκ ∧ o(Dp(s))x, o(Ap)x = dκ ∧ o(Ap(s))x.

Then we have

o∗(Dp)x|Σ(s) ∼ o(Ap(s))x, o∗(Ap)x|Σ(s) ∼ (−1)ind p o(Dp(s))x

o∗(Dp)x ∧ o∗(Ap)x|Σ(s) ∼ o(Dp(s))x ∧ o(Ap(s))x = o(Σ(s))x.

3.2. (Co)orientations of the spaces M2(ξ) and

∫

R

X tY . We define the coori-

entations o∗(M2(ξ))(z,w), o∗(Ap × Dp)(z,w) ∈

∧• T ∗z M ⊗

∧• T ∗wM as follows. If L

is a level surface locus for f , we define

o(L)z = ι(−ξz) o(M)z |L ∈∧2

T ∗z L (z ∈ L, z is not on critical loci).

The space M2(ξ) is the image of the embedding ϕ : M × (0,∞)→ M × M defined

by ϕ(z, T ) = (z,ΦT

−ξ(z)). We fix the orientations of M × M , M ×L and L× M by

o(M × M)(z,w) = o(M)z ∧ o(M)w,

o(M × L)(z,w) = − o(M)z ∧ o(L)w, o(L× M)(z,w) = o(L)z ∧ o(M)w.

If z is not on critical loci and if w = ΦT

−ξ(z), we define

o(M2(ξ))(z,w) = dϕ∗(o(M)z ∧ dT ), o∗(M2(ξ))(z,w) = ∗ o (M2(ξ))(z,w),

o∗(Ap ×Dp)(z,w) = o∗(Ap)z ∧ o∗(Dp)w = ∗(o(Ap)z ∧ o(Dp)w),

where dϕ∗ is the map corresponding to dϕ : T (M × (0,∞)) → T (M × M)¶. The

(co)orientation of

∫

R

X tY is determined by o∗(Ap ×Dp)(z,w).

Let z ∈ M be a point not on any critical loci of ξ and let Uz be a small open

ball around z. Let

M2(ξ;Uz, L) = M2(ξ) ∩ (Uz × L), M2(ξ;L,Uz) = M2(ξ) ∩ (L× Uz).

The former agrees with the image of the embedding ϕ′ : Uz → Uz×L, ϕ′(u) = (u, v),

where v is the intersection point of the gradient line from u with L. The latter

agrees with the restriction of the image of the embedding ϕ′′ : L× (0,∞)→ L×M ,

ϕ′′(u, T ) = (u,ΦT

−ξ(u)). We define

o(M2(ξ; M, L))(z,w) = dϕ′∗(o(M)z), o(M2(ξ;L, M))(z,w) = −dϕ′′

∗(o(L)z ∧ dT ).

These are given so that the coorientations agree with o∗(M2(ξ))(z,w).

¶This may not agree with (ϕ−1)∗, although they are equivalent on∧

4 T ∗(M × (0,∞)).

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 23

3.3. Coorientation of

∫

R

X Ω · · ·Ω tY . Let ~x = (x, z1, z2, . . . , zn, y) be a point of∫

R

X Ω · · ·Ω tY . For a point z ∈ M , let Lz denote a level surface locus of f that

includes z. We identify a neighborhood of ~x in

∫

R

X Ω · · ·Ω tY with a submanifold

of M × Lz1 × · · · × Lzn × M . Let Ai (i = 1, 2, . . . , n + 2) denote the i-th factor

in this product and let αi : M × Lz1 × · · · × Lzn × M → Ai × Ai+1 denote the

projection. The coorientation o∗(Aqi × Dqi) induces coorientation of Aqi × Dqi in

Ai×Ai+1 and we denote the pullback α∗i o

∗(Aqi×Dqi) by o∗(Aqi×Dqi), by abusing

the notation.

Definition 3.1. With these notations, we define the coorienatation of the space∫

R

X Ω · · ·Ω︸ ︷︷ ︸n

tY at ~x by

o∗(Aq1 ×Dq1 )(x,z1) ∧ o∗(Aq2 ×Dq2 )(z1,z2) ∧ · · · ∧ o∗(Aqn+1 ×Dqn+1)(zn,y)

∈ ∧•T ∗(x,z1,...,zn,y)

(M × Lz1 × · · · × Lzn × M)

=∧•

T ∗x M ⊗

∧•T ∗z1Lz1 ⊗ · · · ⊗

∧•T ∗znLzn ⊗

∧•T ∗y M.

The coorientations of

∫

R

X Ω · · ·Ω︸ ︷︷ ︸n

and

∫

R

Ω · · ·Ω︸ ︷︷ ︸n

tY in M × L1 × · · · × Ln and

L1 × · · · × Ln × M are defined respectively by

o∗(Dq1)z1 ∧ o∗(Aq2 ×Dq2)(z1,z2) ∧ · · · ∧ o∗(Aqn+1 ×Dqn+1)(zn,y),

o∗(Aq1 ×Dq1 )(x,z1) ∧ · · · ∧ o∗(Aqn ×Dqn)(zn−1,zn) ∧ o∗(Aqn+1)zn .

We fix the orientations of M × L1 × · · · × Ln × M , M × L1 × · · · × Ln and L1 ×· · ·×Ln× M respectively by o(M)x ∧ o(Lz1)z1 ∧ · · · ∧ o(Lzn)zn ∧ o(M)y, −o(M)x ∧o(Lz1)z1 ∧ · · · ∧ o(Lzn)zn , o(Lz1)z1 ∧ · · · ∧ o(Lzn)zn ∧ o(M)y .

Example 3.2. We consider the coorientation of the iterated integral∫

R

Aq1Mq1q2Mq2q3Dq3

for critical loci q1, q2, q3 of ξ of index 1.

o∗(∫

R

Aq1Mq1q2Mq2q3Dq3

)(x,z1,z2,y)

= o∗(Aq1)x ∧ o∗(Dq1 )z1 ∧ o∗(Aq2)z1 ∧ o∗(Dq2 )z2 ∧ o∗(Aq3)z2 ∧ o∗(Dq3)y

∼ o∗(Aq1)x ∧ ε12 o(Lz1)z1 ∧ ε23 o(Lz2)z2 ∧ o∗(Dq3 )y

= ε12 ε23 o∗(Aq1)x ∧ o(Lz1)z1 ∧ o(Lz2)z2 ∧ o∗(Dq3 )y

(3.3.1)

for the signs ε12, ε23 ∈ −1, 1 of the 1/1-intersections. Then by Definition 3.1, the

coorientation (3.3.1) gives

o(∫

R

Aq1Mq1q2Mq2q3Dq3

)(x,z1,z2,y)

= ε12 ε23 o(Aq1)x ∧ o(Dq3 )y.

24 TADAYUKI WATANABE

3.4. Consistency of the orientations at the boundaries.

Lemma 3.3. The orientations induced from those of M 2(ξ) and

∫

R

X tY on the

codimension 1 stratum S−1∼= X ×R

tY are opposite.

Proof. It suffices to compare the orientations at (x, y) ∈ S−1 induced from those

of M 2(ξ) and

∫

R

A qDq such that both x and y are in a small neighborhood of a

point x0 on a critical locus q of f .

By convention of §3.2,o(M 2(ξ))(x,y) = dϕ∗(o(M)x ∧ dT )

o(∫

R

A qDq

)(x,y)

= o(A q ×Dq)(x,y) = o(Aq)x ∧ o(Dq)y(3.4.1)

We check that these orientations induce opposite orientations at the intersection

stratum S−1.

Case 1: ind q = 1. By the parametrized Morse lemma [Ig1, §A1], there are a

local coordinate system (s, x1, x2) around x0 ∈ M , say on a neighborhood Ux0 , and

a metric on M such that

• x0 corresponds to the origin.

• dκ( ∂∂s) > 0.

• f agrees with h(s, x1, x2) = c(s) − x21

2 +x22

2 on Ux0 for a smooth function

c(s) and that ξ agrees with the gradient of h along the fibers on Ux0 .

Let A = (s, 0, x2); s, x2 ∈ R ∩ Ux0 , D = (s, x1, 0); s, x1 ∈ R ∩ Ux0 . Then

M 2(ξ;Ux0, Ux0) = (s, ρu, v)× (s, u, ρv); s, u, v ∈ R, 0 ≤ ρ ≤ 1 ∩ (Ux0 × Ux0).

This is the image of the embedding ϕ : Ux0 → Ux0 ×Ux0 , ϕ(s, ρ, u, v) = (s, ρu, v)×(s, u, ρv). The boundary of M 2(ξ;Ux0 , Ux0) corresponds to the faces at ρ = 0, 1.

The face at ρ = 0 intersects A × D along S−1. Now we describe the induced

orientation of the face at ρ = 0. Let ds, dx1, dx2 be the standard basis of T ∗xUx0

such that ds dx1dx2 ∼ o(M)x and we take the standard basis of T ∗(x,y)(Ux0 × Ux0)

as follows.

ds = pr∗1ds, ds′ = pr∗2ds, dxi = pr∗1dxi, dyi = pr∗2dxi (i = 1, 2).

We consider a boundary point with u > 0 without loss of generality. The Jacobian

matrix of the mapping (x, T ) 7→ (x,ΦT

−ξ(x)) is as follows (y = ΦT

−ξ(x)).

(I O

dΦT

−ξ−ξy

)

By this and by §3.2, we see that

o(M2(ξ;Ux0 , Ux0))(x,y) = (ds+ ds′)∧ (dx1 + eTdy1)∧ (dx2 + e−Tdy2)∧ (dy1− dy2).With respect to the coordinate given by ϕ, this is equivalent to

(3.4.2) −(ds+ ds′) ∧ (u dx1 + v dy2) ∧ (ρ dx1 + dy1) ∧ (dx2 + ρ dy2).

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 25

Indeed, right multiplication of dy2(−ds+ ds′) to both orientations give

o(M2(ξ;Ux0 , Ux0))(x,y) ∧ dy2 ∧ (−ds+ ds′) = 2 ds dx1 dx2 ds′ dy1 dy2, and

− (ds+ ds′) ∧ (u dx1 + v dy2) ∧ (ρ dx1 + dy1) ∧ (dx2 + ρ dy2) ∧ dy2 ∧ (−ds+ ds′)

= 2u ds dx1 dx2 ds′ dy1 dy2.

The latter expression (3.4.2) makes sense even when (x, y) lies in the boundary.

Since u dx1 + v dy2 is the dual of an inward normal vector to ∂M 2(ξ;Ux0 , Ux0) at

(x, y), we have

(3.4.3) o(∂M 2(ξ;Ux0 , Ux0))(x,y) ∼ −(ds+ ds′) dy1 dx2 (at ρ = 0).

On the other hand, if o(D(s)) = β dx1, then by convention of §3.1, we have

o(A(s)) = β dx2, o(D) = β ds dx1, o(A) = β ds dx2,

o(A)x ∧ o(D)y = ds dx2 ds′ dy1 = −ds ds′ dx2 dy1.

and this gives

o(∂

∫

R

AD)(x,y)

= ι( ∂∂s− ∂

∂s′

)(−ds ds′ dx2 dy1) = (ds+ ds′) dy1 dx2.

This is opposite to (3.4.3).

Case 2: ind q = 2. By the parametrized Morse lemma [Ig1, §A1], there are a

local coordinate system (s, x1, x2) around x0 ∈ M , say on a neighborhood Ux0 , and

a metric on M such that

• x0 corresponds to the origin.

• dκ( ∂∂s) > 0.

• f agrees with h(s, x1, x2) = c(s) − x21

2 −x22

2 on Ux0 for a smooth function

c(s) and that ξ agrees with the gradient of h along the fibers on Ux0 .

Let A = (s, 0, 0); s ∈ R ∩ Ux0 , D = (s, x1, x2); s, x1, x2 ∈ R ∩ Ux0 . Then

M 2(ξ;Ux0, Ux0) = (s, ρu, ρv)× (s, u, v); s, u, v ∈ R, 0 ≤ ρ ≤ 1 ∩ (Ux0 × Ux0).

This is the image of the embedding ϕ : Ux0 → Ux0×Ux0 , ϕ(s, ρ, u, v) = (s, ρu, ρv)×(s, u, v). The boundary of M 2(ξ;Ux0 , Ux0) corresponds to the faces at ρ = 0, 1. The

face at ρ = 0 intersects A×D along S−1. Now we describe the induced orientation

of the face at ρ = 0, u > 0. Let ds, dx1, dx2, ds′, dy1, dy2 be the standard basis of

T ∗(x,y)(Ux0 × Ux0) as above. By §3.2, we see that if x is not on the critical loci of ξ

and if y = ΦT

−ξ(x),

o(M2(ξ;Ux0 , Ux0))(x,y) = (ds+ ds′) ∧ (dx1 + eTdy1) ∧ (dx2 + eTdy2) ∧ (dy1 + dy2).

With respect to the coordinate given by ϕ, this is equivalent to

(3.4.4) −(ds+ ds′) ∧ (u dx1 + v dx2) ∧ (ρ dx1 + dy1) ∧ (ρ dx2 + dy2).

Indeed, right multiplication of dy2(−ds+ ds′) to both orientations give

o(M2(ξ;Ux0 , Ux0))(x,y) ∧ dy2 ∧ (−ds+ ds′) = 2 ds dx1 dx2 ds′ dy1 dy2, and

− (ds+ ds′) ∧ (u dx1 + v dx2) ∧ (ρ dx1 + dy1) ∧ (ρ dx2 + dy2) ∧ dy2 ∧ (−ds+ ds′)

= 2uρ ds dx1 dx2 ds′ dy1 dy2.

26 TADAYUKI WATANABE

The latter expression (3.4.4) makes sense even when (x, y) lies in the boundary.

Since u dx1 + v dx2 is the dual of an inward normal vector to ∂M 2(ξ;Ux0 , Ux0) at

(x, y), we have

(3.4.5) o(∂M 2(ξ;Ux0 , Ux0))(x,y) ∼ −(ds+ ds′) dy1 dy2 (at ρ = 0).

On the other hand, if o(D(s)) = β dx1 dx2, then by convention of §3.1, we have

o(A(s)) = β, o(D) = β ds dx1 dx2, o(A) = β ds,

o(A)x ∧ o(D)y = ds ds′ dy1 dy2.

and this gives

o(∂

∫

R

AD)(x,y)

= ι( ∂∂s− ∂

∂s′

)ds ds′ dy1 dy2 = (ds+ ds′) dy1 dy2.

This is opposite to (3.4.5).

The case where ind q = 0 is the same as the case where ind q = 2. This completes

the proof.

Lemma 3.4. The orientations induced from those of X and

∫

R

X Ω on the codi-

mension 1 stratum X ×R Ω are opposite.

Proof. It suffices to check that A q and

∫

R

A pM pq induce opposite orientations at

(z, w) ∈ A p×RM pq for critical loci p, q. Let x0 be a point of p. By the parametrized

Morse lemma [Ig1, §A1], there are a local coordinate system (s, x1, x2) around

x0 ∈ M , say on a neighborhood Ux0 , and a metric on M such that x0 corresponds

to the origin and f agrees with

h(s, x1, x2) =

c(s) +

x21

2 −x22

2 if ind p = 1

c(s)− x21

2 −x22

2 if ind p = 2

on Ux0 for a smooth function c(s) and that ξ agrees with the gradient of h along

the fibers on Ux0 . Then by convention of §3.2,

o∗(Dp) =

−β dx1 if ind p = 1

β if ind p = 2, o∗(Ap) =

−β dx2 if ind p = 1

β dx1dx2 if ind p = 2

for some β ∈ −1, 1. In the coorinate system above, we put z = (s, z1, z2),

w = (s′, w1, w2). We assume without loss of generality that Lw is perpendicular to

ξ at w. We check the assertion for the possible pairs (ind p, ind q) = (2, 1), (1, 0),

(1, 1), (2, 0).

Case 1: (ind p, ind q) = (2, 1). Suppose that z = (0, 0, 0) and that w lies in the

negative part of the x2-axis. Let Uz be a small open ball around z. We assume that

Aq locally agrees with (s, 0, x2) ; x2 < 0. We may put o(Aq) = αds dx2 for some

α ∈ −1, 1. The point z lies in the image of ∂A q under b, which locally agrees

with the s-axis. We identify a neighborhood of b−1(z) in A q with its image under

b. Thus o(A q)z makes sense under this identification. Now we have

o(∂A q)z = ι( ∂

∂z2

)α ds dz2 = −αds.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 27

The induced orientation of the image of the embedding A q ∩ Uz → M × Lw is

(3.4.6) −α(ds+ ds′).

On the other hand, by Definition 3.1, the coorientation of Ap×Mpq in M ×Lw

at (z, w) is

o∗(Ap)z ∧ o∗(Dp)w ∧ o∗(Aq)w = β dz1 dz2 ∧ β ∧ (−α dw1) = −αdz1 dz2 dw1.

According to the convention of §3.2, we have o(M ×Lw)(z,w) = −ds dz1 dz2 ds′ dw1

and we have

o(Ap ×Mpq)(z,w) = αds ds′.

This induces

o(∂

∫

R

A pM pq

)(z,w)

= ι( ∂∂s− ∂

∂s′

)αds ds′ = α(ds + ds′).

This is opposite to (3.4.6).

Case 2: (ind p, ind q) = (1, 0). Suppose that z lies in the x1-axis and that w lies

in the negative part of the x2-axis. Let Uz be a small ball around z. We assume

that Aq locally agrees with (s, x1, x2) ; x2 < 0. We may put o(Aq) = αds dx1 dx2for some α ∈ −1, 1. This gives

o(∂A q)z = ι( ∂

∂z2

)αds dz1 dz2 = αds dz1.

The induced orientation of the image of the embedding A q ∩ Uz → M × Lw is

(3.4.7) α(ds+ ds′) dz1.

On the other hand, the coorientation of Ap ×Mpq in M × Lw at (z, w) is

o∗(Ap)z ∧ o∗(Dp)w ∧ o∗(Aq)w = (−β dz2) ∧ (−β dw1) ∧ α = αdz2dw1.

According to the convention of §3.2, we have o(M ×Lw)(z,w) = −ds dz1 dz2 ds′ dw1

and we have

o(Ap ×Mpq)(z,w) = −αds ds′ dz1.This induces

o(∂

∫

R

A pM pq

)(z,w)

= ι( ∂∂s− ∂

∂s′

)(−αds ds′ dz1) = −α(ds+ ds′) dz1.

This is opposite to (3.4.7).

Case 3: (ind p, ind q) = (1, 1). Suppose that z = (0, 0, 0) and that w = (0, 0,−1).The intersection of Aq with the plane x2 = −1 agrees with the set of points

(s, µs,−1) for a constant µ 6= 0. Hence Aq agrees locally with the set of points

(s, µseT ,−e−T ). By setting a′1 = µseT , s = −s/a′1, we have

(s, µseT ,−e−T ) = (−sa′1, a′1, sµ).The right hand side makes sense for s = 0, a′1 = 0. Thus the right hand side

gives the local coordinate system for A q around z. Since the Jacobian matrix for

(s, a′1) 7→ (−sa′1, a′1, sµ) is (−a′

1−s

0 1

µ 0

),

28 TADAYUKI WATANABE

we may put o(Aq)z = αµdz1 dz2 for some α ∈ −1, 1. This gives

(3.4.8) o(∂A q)z = ι( ∂

∂z2

)αµdz1 dz2 = −αµdz1.

On the other hand, the coorientation of Ap ×Mpq in M × Lw at (z, w) is

o∗(Ap)z ∧ o∗(Dp)w ∧ o∗(Aq)w = −β dz2 ∧ (−β dw1) ∧ αµds′ = αµdz2dw1ds′.

According to the convention of §3.2, we have o(M ×Lw)(z,w) = −ds dz1 dz2 ds′ dw1

and we have

o(Ap ×Mpq)(z,w) = αµds dz1.

This induces

o(∂

∫

R

A pM pq

)(z,w)

= ι( ∂∂s

)αµds dz1 = αµdz1.

This is opposite to (3.4.8).

Case 4: (ind p, ind q) = (2, 0). Suppose that z = (0, 0, 0) and that w =

(0, 0,−1). We may assume that Aq locally agrees with R3 − p. We may put

o(Aq) = αds dx1 dx2 for some α ∈ −1, 1. Let Uz be a small open neighborhood

of z in M and let L′w = (s′, w1,−1) ; s′, w1 ∈ R. Then Aq ∩ Uz − 0 can be

identified with the space

M2(ξ;Uz, L′w) = M2(ξ) ∩ (Uz × L′

w)

= (s, ρu,−ρ)× (s, u,−1) ; s, u ∈ R, ρ ∈ [0, 1] ∩ (Uz × L′w).

By convention of §3.2, we have

o(M2(ξ;Uz, L′w))(z,w) = (ds+ ds′) ∧ (dz1 + eTdw1) ∧ dz2

for some T > 0. Hence o(Aq) induces the following orientation of M2(ξ;Uz, L′w):

α (ds+ ds′) ∧ (dz1 + eTdw1) ∧ dz2.Since the Jacobian matrix of the embedding (s, ρ, u) 7→ (s, ρu,−ρ)× (s, u,−1) is

1 0 0

0 u ρ

0 −1 0

1 0 0

0 0 1

0 0 0

,

the orientation of o(M2(ξ;Uz, L′w))(z,w) is a multiple of (ds+ ds′)∧ (u dz1− dz2)∧

(ρ dz1 + dw1). In fact, the induced orientation from o(Aq) is equivalent to

α (ds+ ds′) ∧ (u dz1 − dz2) ∧ (ρ dz1 + dw1).

The equivalence can be checked by multiplying dw1 dw2 (−ds + ds′) to both ori-

entations. When ρ = 0, u dz1 − dz2 is the dual of the inward normal vector of

M2(ξ;Uz, L2) at (z, w). We see that the induced orientation on the boundary is

(3.4.9) α(ds + ds′) dw1.

On the other hand, the coorientation of Ap ×Mpq in M × L′w at (z, w) is

o∗(Ap)z ∧ o∗(Dp)w ∧ o∗(Aq)w = β dz1 dz2 ∧ β ∧ α = αdz1 dz2.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 29

According to the convention of §3.2, we have o(M ×L′w)(z,w) = −ds dz1 dz2 ds′ dw1

and we have

o(Ap ×Mpq)(z,w) = −αds ds′ dw1.

This induces

o(∂

∫

R

A pM pq

)(z,w)

= ι( ∂∂s− ∂

∂s′

)(−α ds ds′ dw1) = −α(ds+ ds′) dw1.

This is opposite to (3.4.9).

Lemma 3.5. The orientations induced from those of tY and

∫

R

Ω tY on the codi-

mension 1 stratum Ω×RtY are opposite.

Lemma 3.5 can be proved by a similar argument as in the proof of Lemma 3.4.

Corollary 3.6. The orientations induced from those of Ω and

∫

R

ΩΩ on the codi-

mension 1 stratum Ω×R Ω are opposite.

Proof. It suffices to check that M pr and

∫

R

M pqM qr induce opposite orientations

at M pq ×R M qr. By convention,

o∗(Mpr)z = o∗(Dp)z ∧ o∗(Ar)z ,

o∗(∫

R

MpqMqr

)(z,w)

= o∗(Dp)z ∧ o∗(Aq)z ∧ o∗(Dq)w ∧ o∗(Ar)w.

According to Lemma 3.4, the coorientations o∗(Ar)z and o∗(Aq)z ∧ o∗(Dq)w ∧o∗(Ar)w = o∗(

∫R

AqMqr)(z,w) induce opposite orientations at the common bound-

ary. This completes the proof.

Corollary 3.7. For n ≥ 0, the orientations induced from those of

∫

R

X Ω · · ·Ω︸ ︷︷ ︸n

tY

and

∫

R

X Ω · · ·Ω︸ ︷︷ ︸n+1

tY on the codimension 1 stratum Sn = Tn+1 are opposite.

Proof. According to Definition 3.1, the induced coorientations at each point of the

codimension 1 stratum are of the forms A∧P ∧B and A∧Q∧B respectively, where

(P,Q) is the pair that has been proved to be opposite to each other in Lemmas 3.4,

3.5 and Corollary 3.6. This completes the proof.

4. A chain from the moduli space of Z-paths

In this section, we shall show that the natural map from MZ

2 (ξ)Z to M ×Z M

gives a 4-dimensional Q(t)-chain P (ξ) (Lemma 4.2). We consider the blow-up of

P (ξ) along the lifts of the diagonal in M ×Z M . The result is a 4-dimensional Q(t)-

chain Q(ξ) in the equivariant configuration space C2,Z(M) and we give an explicit

formula for ∂Q(ξ) (Theorem 4.7). We show that Q(ξ) is in a sense an explicit

representative for Lescop’s equivariant propagator (Corollary 4.11).

30 TADAYUKI WATANABE

4.1. Signs of Z-paths. Here we define the signs of Z-paths with fixed endpoints.

Let Σ = κ−1(0). In the following, we assume that there are no 1/1-intersections in

Σ, without loss of generality. Let p, q be critical loci of ξ of the same index that

intersect Σ at p0, q0 respectively. The space

MZ

2 (ξ; p0, tiq0) = M

Z

2 (ξ) ∩ b−1(p0 × tiq0)

is a compact oriented 0-manifold. Thus the natural map b : MZ

2 (ξ; p0, tiq0) →

p0× tiq0 is a finite covering map and represents a 0-dimensional chain in M ×M ,

which can be written as n · p0 ⊗ tiq0 for an integer n. The integer n is determined

as follows. As we have seen in Example 3.2, the coorientation of MZ

2 (ξ; p0, tiq0)

considered in p0 × L1 × · · · × Lm−1 × tiq0 is given by

ε1 o(L1)z1 ∧ ε2 o(L2)z2 ∧ · · · ∧ εr o(Lr)zr

for the signs εi ∈ −1, 1. We define the sign of the 0-chain b(p0, z1, z2, . . . , zr, tiq0)

as ε1ε2 · · · εr. Then the integer n is determined as the sum of the signs of all the

0-simplices over p0 × tiq0.

4.2. Action of Z-paths on Morse complexs. Here, we assume that all the

homology groups are considered with coefficients in Q. ConsiderM as the mapping

torus of an orientation preserving diffeomorphism ϕ : Σ → Σ, where Σ = κ−1(0).

The restriction of the fiberwise gradient ξ to Σ defines a handle filtration ∅ =

Σ(−1) ⊂ Σ(0) ⊂ Σ(1) ⊂ Σ(2) = Σ.

Put Ci(Σ) = Hi(Σ(i),Σ(i−1)). Then ϕ induces endomorphisms ϕ♯i : Ci(Σ) →

Ci(Σ) and ϕ∗i : Hi(Σ) → Hi(Σ). Note that ϕ♯i is uniquely determined for ξ as

follows. ϕ♯0 and ϕ♯2 are induced by the permutation given by the graphic. It can

be seen by using the Morse lemma that a 1/1-intersection corresponds to a slide

of a 1-handle over another 1-handle, as is well known (e.g. [Mi, Theorem 7.6]).

Thus ϕ♯1 can be seen as the induced map on H1 of the corresponding base pointed

homotopy equivalence∨S1 → ∨

S1, which is uniquely determined by the sequence

of 1-handle slides for the fiberwise gradient ξ. Namely, if a 1-handle α slides over

another one β, then let [α] mapped to [α]± [β] and fix other handles.

Now we shall see that ϕ♯i is also obtained by counting Z-paths. The cases i = 0, 2

are obvious. For the case i = 1, let c1, c2, . . . , cN be the critical points of ξ|Σ of

index 1. We identify C1(Σ) with the vector space generated by c1, . . . , cN. We

define an endomorphism Θ(ξ) : C1(Σ)→ C1(Σ) by

Θ(ξ)(ci) =N∑

j=1

N(ci, tcj) cj ,

where N(ci, tcj) ∈ Z is the count of Z-paths in M from ci to tcj counted with signs

as in §4.1.

Lemma 4.1. ϕ♯1 = Θ(ξ).

Proof. Let u1, . . . , ur ∈ S1 be the level exchange parameters considered in §2.1.Choose a small number ε > 0 and let J2j−1 = [uj − ε, uj + ε] and J2j = [uj +

ε, uj+1 − ε]. Then we have S1 =⋃r

j=1 Jj . The both sides of Lemma 4.1 can

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 31

Figure 7

be decomposed as the compositions of corresponding morphisms ϕJj

♯1 and ΘJj(ξ) :

H1(Σ(1)(aj),Σ

(0)(aj)) → H1(Σ(1)(bj),Σ

(0)(bj)) on Jj = [aj , bj ] from left to right.

We check that ϕJj

♯1 and ΘJj(ξ) coincide for each j.

If j is odd, then there are no 1/1-intersections in κ−1(Jj). In this case ϕJj

♯1 and

ΘJj(ξ) are given by the same permutation on the set of the critical points of f |Σ(aj),

totally ordered by the value of f |Σ(aj). So we need only to consider the case that

j is even. By decomposing Jj by smaller intervals Jjk = [ajk, bjk] each containing

just one 1/1-intersection, it suffices to check Lemma 4.1 for the case when there is

only one 1/1-intersection and there is no level exchange bifurcations. There are two

possibilities for a handle-sliding, as shown in Figure 7. Moreover, there are four

possibilities for the coorientations of the two descending manifold loci that are the

cores of the two 1-handles.

We mainly consider the first case in Figure 7 as the second case is similar to the

first one. Let p, q be critical loci of ξ|MJjkand suppose that Dp slides over Dq in

ξ|MJjk, as s increases from ajk to bjk. Let d ∈ p, d′ ∈ q be the endpoints of the

flow-line of ξ|MJjkfrom p to q and put cjk = κ(d) = κ(d′). For a critical locus v

of ξ|MJjk, we write v0 = v(ajk) and v1 = v(bjk) here. Choose a local coordinate

system (s, x1, x2) around d′ such that

• the x1x2-plane agrees with the level surface Σ(cjk) = κ−1(cjk) of κ at d′,

• dκ( ∂∂s) > 0,

• the sx1-plane agrees with Dq,

• the sx2-plane agrees with Aq.

We only consider one special case about the orientations of descending manifolds of

p and q out of the four since the other cases can be checked by the same argument

as the special case. So we assume the following, applying the convention in §3.1.o(Dq(cjk))x = dx1, o(Dq)x = −ds ∧ dx1,o(Dp(cjk))x = −dx2, o(Dp)x = −(ds+ µ dx1) ∧ (−dx2)

for a real number µ > 0 and for x on the flow-line dd′ that is close to d′. See

Figure 8. This gives

o∗(Dq)x = −dx2, o∗(Dp)x = µ ds− dx1, and

o∗(Dp)x ∧ o∗(Aq)x = (µ ds− dx1) ∧ (−dx1) = −µ ds ∧ dx1 = −µ o(L)x,where L is the level surface locus of f in κ−1(Jjk) including x. From this and §4.1,we have, for each critical locus v of ξ|MJjk

,

ΘJjk(ξ)(v0) =

p1 −

µ

|µ|q1 if v0 = p0

v1 other critical points

32 TADAYUKI WATANABE

Figure 8

This agrees with the homological action ϕJjk

♯1 of the homotopy equivalence ϕJjk :

Σ(1)(ajk)/Σ(0)(ajk)→ Σ(1)(bjk)/Σ

(0)(bjk). More precisely, for the 1-handles hp, hq, h′p, h

′q

in Figure 8, we have

ϕJjk

♯1 ([hp]) =

[h′p]− [h′q] if µ > 0

[h′p] + [h′q] if µ < 0

in H1(Σ(1)(bjk),Σ

(0)(bjk)). If one of o(Dp) and o(Dq) is reversed, then both ϕJjk

♯1

and ΘJjk(ξ) are reversed. This completes the proof of ϕJjk

♯1 = ΘJjk(ξ).

4.3. Making MZ

2 (ξ)Z into a Q(t)-chain in M ×Z M .

Lemma 4.2. The natural map b : MZ

2 (ξ)Z → M ×Z M gives a 4-dimensional

Q(t)-chain. (We denote the Q(t)-chain by P (ξ).) Moreover, if b1(M) = 1, then for

a product C(t) of cyclotomic polynomials, C(t)∆(M)P (ξ) is a Λ-chain.

Proof. If a horizontal segment σ in a Z-path is on a critical locus of f of index i,

then we say that σ has index i. Let Gmn be the subspace of M Z2 (ξ) consisting of

Z-paths such that the numbers of horizontal segments of indices 0, 1, 2 are m, 0,

n respectively. Let Hmn be the subspace of M Z2 (ξ) consisting of Z-paths such that

the numbers of horizontal segments of indices 0, 2 are m, n respectively and that

has at least one horizontal segment of index 1. Since a Z-path can visit critical loci

of index 2 only once and also that of index 0 only once, we have

MZ2 (ξ) =

∑

(m,n)

(Gmn +Hmn),

where (m,n) ∈ (0, 0), (1, 0), (0, 1), (1, 1). Let Gmn and Hmn be the closures of

Gmn and Hmn respectively, in MZ

2 (ξ). Since Hmn and Gmn are invariant under

the diagonal Z-action on MZ

2 (ξ), we have the quotients (Hmn)Z and (Gmn)Z by

the Z-action. It suffices to check that the restrictions of b to (Hmn)Z and (Gmn)Zare Q(t)-chains. This is checked in Lemmas 4.3 and 4.4 below.

Lemma 4.3. Hmn is a countable union of smooth compact manifolds with corners.

The natural map (Hmn)Z → M×ZM gives a 4-dimensional Q(t)-chain in M×ZM .

Moreover, if b1(M) = 1, then for a product C(t) of cyclotomic polynomials, the

multiplication by C(t)∆(M) of the Q(t)-chain yields a Λ-chain.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 33

Proof. We consider the decomposition M =⋃

i∈ZM [i] where M [i] = κ−1[i, i + 1].

Let Σ[i] = κ−1(i) for i ∈ Z. First we consider the simplest case (m,n) = (0, 0).

Suppose that k > j. Let H00(M [k],M [j]) be the subspace of H00 consisting of

Z-paths from points of M [k] to points of M [j]. The critical loci of f intersects

Σ[k] transversally at finitely many points. Let x1, x2, . . . , xr ∈ Σ[k] be all the

intersection points with critical loci of index 1 and let yi = tk−j−1xi ∈ Σ[j + 1].

Let v1, . . . , vr ⊂ M be the critical loci of f of index 1 that intersect x1, x2, . . . , xrrespectively and let cℓ1 =

⋃rj=1 vj . Then H00(M [k],M [j]) can be written as

H00(M [k],M [j]) =∑

1≤α,β≤r

H00(M [k],M [j])αβ,

where H00(M [k],M [j])αβ is the subspace of H00(M [k],M [j]) consisting of Z-paths

γ such that γ ∩ cℓ1 ∩ Σ[k] = xα and γ ∩ cℓ1 ∩ Σ[j + 1] = yβ. Note that we are

assuming that there are no 1/1-intersection on Σ[i] for all i ∈ Z. Let H0∗(M [k])α

be the subspace of MZ

2 (ξ) consisting of Z-paths from points in M [k] to xα. Let

H∗0(M [j])β be the subspace of MZ

2 (ξ) consisting of Z-paths from yβ to points in

M [j].

Now we decompose the evaluation map (H00)Z → M ×Z M as a formal sum

of smooth maps from the compact pieces H00(M [k],M [j])αβ. There is a natural

homeomorphism

ωαβ(k, j) : H00(M [k],M [j])αβ≈→ H0∗(M [k])α × Iαβ(k, j)×H∗0(M [j])β ,

where Iαβ(k, j) is the moduli space of Z-paths from xα to yβ . The evaluation

map evαβ(k, j) : Iαβ(k, j) → xα × yβ is a finite covering map. We define pr1 :

H0∗(M [k])α →M [k] and pr2 : H∗0(M [j])β →M [j] as

pr1(γ1) = σ1(µ1), pr2(γ2) = σ′N ′(ν′N ′),

where γ1 = (σ1, . . . , σN ) ∈ H0∗(M [k])α, γ2 = (σ′1, . . . , σ

′N ′) ∈ H∗0(M [j])β , σi :

[µi, νi]→M and σ′j : [µ

′j , ν

′j ]→M . Let ι :M [k]×xα×yβ×M [j]→M [k]×M [j]

be the projection map. The natural map

ϕαβ(k, j) : H00(M [k],M [j])αβ →M [k]×M [j]

induced by b is factorized as ϕαβ(k, j) = ι(pr1×evαβ(k, j)×pr2)ωαβ(k, j). Then

ϕαβ(k, j) can be considered as a chain inM [k]×M [j]. We define the 4-dimensional

chain Φ(k, j) in M [k]×M [j] by

Φ(k, j) =∑

1≤α,β≤r

ϕαβ(k, j),

which can be represented by the evaluation map H00(M [k],M [j])→M [k]×M [j],

which gives the endpoints of paths. We consider the formal sum

Φ =

∞∑

n=0

Φ(n, 0),

which can be represented by the evaluation map (H00)Z → M ×Z M .

We check that Φ is well-defined as a Q(t)-chain, by an analogous argument

as [Pa1]. Let Uα[k] be the 2-dimensional chain in M [k] represented by the first

34 TADAYUKI WATANABE

projection pr1 : H0∗(M [k])α → M [k] and let Vβ [j] be the 2-dimensional chain in

M [j] represented by the second projection pr2 : H∗0(M [j])β → M [j]. Let nαβ be

the integer determined by the equation

evαβ(2, 0)♯(Iαβ(2, 0)) = nαβ(xα ⊗ yβ),along the convention in §4.1. Let A1 denote the matrix (nαβ), which represents ϕ♯1.

For a matrix X , we denote by Xαβ the (α, β)-entry of X .As a chain in M ×Z M ,

we have

ϕαβ(n, 0) = (An−11 )αβUα[n]⊗ Vβ [0]

= (An−11 )αβt

n−1(Uα[1]⊗ Vβ [0]) = (tA1)n−1αβ Uα[1]⊗ Vβ [0]

when n ≥ 1. Therefore,

Φ(n, 0) =∑

1≤α,β≤r

(tA1)n−1αβ Uα[1]⊗ Vβ [0] (for n ≥ 1),

∞∑

n=0

Φ(n, 0) = Φ(0, 0) +∑

1≤α,β≤r

(1− tA1)−1αβUα[1]⊗ Vβ [0].

Since (1− tA1)−1 is a matrix with entries in Q(t) and Φ(0, 0) is compact, this shows

that Φ represents a Q(t)-chain in M ×Z M . The coefficient (1 − tA1)−1αβ is of the

form B(t) det(1− tA1)−1 for some B(t) ∈ Λ. If b1(M) = 1, then by Lemma 4.1 and

by the identities ∆(M) = c t−g(Σ) det(1 − tϕ∗1) and∏2

i=0 det(1 − tϕ∗i)(−1)i+1

=∏2i=0 det(1− tϕ♯i)

(−1)i+1

, we have

det(1 − tA1)−1 =

c t−g(Σ)(1 − t)2∆(M) det(1− tϕ♯0) det(1− tϕ♯2)

.

Since ϕ♯0 and ϕ♯2 are permutations, det(1 − tϕ♯0) det(1 − tϕ♯2) is a product of

polynomials of the form 1− tk. Hence the denominator is removed if multiplied by

C(t)∆(M) for a product C(t) of cyclotomic polynomials.

For (m,n) = (1, 0), let H00(wi, M) be the subspace of H00 consisting of Z-paths

from points of a critical locus wi of index 2 to M . Let H1∗(wi, wi) be the subspace

of H10 consisting of Z-paths between two points in wi. Note that a sequence in

H1∗(wi, wi) consists of only the endpoints since the index of wi is 2. From the result

for H00 above, we see that the restrictions of b to H00(wi, M)Z and H1∗(wi, wi)Zgive Q(t)-chains in wi ×Z M and wi ×Z wi respectively. Then (H10)Z is the image

of the projection from the fiber product

H1∗(wi, wi)×wiH00(wi, M).

The restriction of b to (H10)Z gives a Q(t)-chain. Indeed, the fiber product is

identified with1

1− t−kwi ×H00(wi, M),

where wi = π(wi) and k is the period of wi. This identification is Z-equivariant

with respect to the integer shifts of the endpoint on wi. The case (m,n) = (0, 1) is

symmetric to this case. If b1(M) = 1, then a multiplication by (1− t−k)C(t)∆(M)

removes the denominator.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 35

Figure 9

For (m,n) = (1, 1), let H00(wi, wj) be the subspace of H00 consisting of Z-paths

from a point in a critical locus wi of index 2 to a point in a critical locus wj of

index 0. Then (H11)Z is the image of the projection from the fiber product

H1∗(wi, wi)×wiH00(wi, wj)×wj

H∗1(wj , wj).

The restriction of b to the image gives a Q(t)-chain. This is Z-equivariantly iden-

tified with1

1− t−kγi ×H00(wi, wj)×

1

1− tℓ γi.

If b1(M) = 1, then a multiplication by (1 − t−k)(1 − tℓ)C(t)∆(M) removes the

denominator.

Lemma 4.4. Gmn is a countable union of smooth compact manifolds with corners.

The natural map b : (Gmn)Z → M ×Z M gives a 4-dimensional Q(t)-chain in M ×Z

M . Moreover, if b1(M) = 1, then for a product C(t) of cyclotomic polynomials, the

multiplication by C(t)∆(M) of the Q(t)-chain yields a Λ-chain.

Proof. The proof is almost parallel to Lemma 4.3. We need only to check that G00,

the one that intersects ∆M, is the union of manifolds with corners. By Lemma 2.5,

it suffices to study only the piecewise smooth structure near ∆M, in particular,

near the diagonal set of a critical locus. We consider only a small neighborhood

of a critical locus of index 1 since the cases of other indices are easier than this.

By the parametrized Morse lemma [Ig1, §A1], it suffices to consider the trivial 1-

parameter family of standard Morse functions h : R2 → R, h(x1, x2) = −x21

2 +x22

2 .

For simplicity, we assume that ξ is its gradient with respect to the Euclidean metric,

without loss of generality. By Lemma 2.9, the closure of the moduli space of

flow lines M2(h) of h is the image of the map ρ : [0, 1] × R1 × R1 → R2 × R2,

ρ(η, a, b) = (ηa, b)× (a, ηb). Let A be the image of ρ. Let

A′ = (a, b, η′); a, b ∈ R1, η′ ∈ [0,√a2 + b2].

Then the mapping (ηa, b) × (a, ηb) 7→ (a, b, η√a2 + b2) defines a homeomorphism

α : A → A′, which is smooth except for the origin. One may see that A′ is the

union of four manifolds with corners, as in Figure 9. This completes the proof.

Lemma 4.5. The boundary of the Q(t)-chain P (ξ) concentrates on the lift of ∆M .

Proof. By Lemma 2.6, the strata

∫

R

X Ω · · ·Ω︸ ︷︷ ︸n

tY and

∫

R

X Ω · · ·Ω︸ ︷︷ ︸n+1

tY for n ≥ 0 are

glued together along the codimension 1 stratum Sn = Tn+1 and by Corollary 3.7,

36 TADAYUKI WATANABE

the boundaries cancel each other out. Hence the boundary of MZ

2 (ξ) concentrates

on the lift of ∆M .

4.4. Blow-up along the diagonal.

Definition 4.6. LetBℓb−1(∆M)(M

Z

2 (ξ)Z) be the blow-up of MZ

2 (ξ)Z along b−1(∆M ),

i.e., the union of the blow-ups of the manifold strata. Let Q(ξ) be the 4-dimensional

Q(t)-chain in C2,Z(M) represented by the natural map Bℓb−1(∆M)(M

Z

2 (ξ)Z) →C2,Z(M) induced by b.

We say that a nonconstant Z-path γ in M is a Z-cycle if b(γ) ∈ ∆M . In other

words, if the endpoints of γ are x and y, then γ is a Z-cycle if moreover π(x) = π(y).

A Z-cycle γ descends to a piecewise smooth map γ : S1 → M , which can be

considered as a “closed orbit” in M . We will also call γ a Z-cycle. A Z-cycle has

an orientation that is determined by the orientations of descending and ascending

manifolds loci of ξ as in §4.1. Then we define the sign ε(γ) ∈ −1, 1 and the period

p(γ) of γ by the following equation

p(γ) = |〈[dκ], [γ]〉|, ε(γ) =〈[dκ], [γ]〉|〈[dκ], [γ]〉| .

If b1(M) = 1 and if K is a knot in M such that 〈[dκ], [K]〉 = 1, then

[γ] = ε(γ) p(γ)[K] ∈ H1(M ;Q).

Let ST (γ) be the pullback γ∗ST (M), which can be considered as a piecewise smooth

3-dimensional chain in ∂C2,Z(M). We say that two Z-cycles γ1 and γ2 are equivalent

if there is a degree 1 homeomorphism g : S1 → S1 such that γ1g = γ2. The indices

of horizontal segments in a Z-cycle must be all equal since a Z-path is descending.

We define the index of a Z-cycle γ to be the index of a horizontal segment in γ.

Let M0 = M \⋃γ:critical locus γ and let sξ : M0 → ST (M0) be the normalization

−ξ/‖ξ‖ of the section −ξ. The closure sξ(M0) in ST (M) is a manifold with bound-

ary whose boundary is the disjoint union of circle bundles over the critical loci γ

of ξ, for a similar reason as [Sh, Lemma 4.3]. The fibers of the circle bundles are

equators of the fibers of ST (γ). Let E−γ be the total space of the 2-disk bundle over

γ whose fibers are the lower hemispheres of the fibers of ST (γ) which lie below (the

tangent space of) the level surfaces of κ. Then ∂sξ(M0) =⋃

γ ∂E−γ as sets. Let

s∗ξ(M) = sξ(M0) +⋃

γ

E−γ ⊂ ST (M).

This is a 3-dimensional piecewise manifold. We orient s∗ξ(M) by extending the

natural orientation (s−1ξ )∗o(M) on sξ(M0) induced from the orientation o(M) of

M . The piecewise smooth projection s∗ξ(M) → M is a homotopy equivalence and

s∗ξ(M) is homotopic to sξ.

Let us fix an orientation of ST (M) and ST (γ). Recall that Bℓ0(R3) can be

identified with the closure in S2 × R3 of the image of the section s : R3 \ 0 →S2 × (R3 \ 0), s(x) = ( x

‖x‖ , x). Let pr1 : S2 × (R3 \ 0) → S2 and pr2 :

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 37

S2 × (R3 \ 0) → R3 \ 0 be the projections. Let ω be the closed 2-form on

R3 \ 0 given by

ω(x) =1

‖x‖3 (x1 dx2 ∧ dx3 − x2 dx1 ∧ dx3 + x3 dx1 ∧ dx2).

Then the pullback pr∗1ω agrees on Im s with pr∗2ω. This shows that pr∗1ω can

be smoothly extended over Bℓ0(R3) by pr∗2ω. We define the standard orientation

o(∆M ) of ∆M as follows. If T ∗xM is spanned by e1, e2, e3 and if o(M)x = e1∧e2∧e3,

then T ∗(x,x)∆M ⊂ T ∗

xM⊕T ∗xM is spanned by pr∗1ei+pr∗2ei, i = 1, 2, 3, and we define

o(∆M )(x,x) =∧3

i=1(pr∗1ei+pr∗2ei). Then we have o∗(∆M )(x,x) =

∧3i=1(pr

∗2ei−pr∗1ei)

and this induces an orientation of the fibers of N∆M. Now we orient ST (M) by the

smooth extension of

φ∗(pr∗1ω ∧ o(∆M )),

where φ is the trivialization in (1.3.1) and pr1 : R3 \ 0 ×∆M → R3 \ 0 is theprojection. It is easy to check that φ∗(pr∗1ω ∧ o(∆M )) is equivalent to the smooth

extension of ι(n) o(∆M ) ∧ o∗(∆M ) = o(∆M ) ∧ ι(−n) o∗(∆M ) in Bℓ0(N∆M), where

n is a vector field on N∆M\∆M along the fibers that points toward ∆M . Similarly,

we orient ST (γ) for a Z-cycle γ by

φ∗(pr∗1ω ∧ o(∆γ)),

where o(∆γ) = pr∗1o(γ) + pr∗2o(γ) and o(γ) = κ∗ds.

For a Z-cycle γ, we denote by γirr the minimal Z-cycle such that γ is equivalent

to the iteration (γirr)k for a positive integer k and we call γirr the irreducible factor

of γ. This is unique up to equivalence.

Theorem 4.7. The boundary of the 4-dimensional Q(t)-chain Q(ξ) is given by

∂Q(ξ) = s∗ξ(M) +∑

γ

(−1)indγε(γ) tp(γ) ST (γirr),

where the sum is taken over equivalence classes of Z-cycles in M . Moreover, if

b1(M) = 1, then for a product C(t) of cyclotomic polynomials, C(t)∆(M)Q(ξ) is a

Λ-chain.

Lemma 4.8. The face of ∂Q(ξ) at a Z-cycle γ without vertical segments contributes

as (−1)indγtp(γ)ST (γirr).

Proof. Suppose that γ is a Z-cycle from x0 to tix0 without vertical segments, where

i = p(γ). Here we identify N∆Mwith a small tubular neighborhood of ∆M inM×M

in a usual way.

Case 1: ind γ = 1. Let x0, Ux0 , A, D be as in Case 1 of the proof of Lemma 3.3.

Then we have∫

R

X tY ∩ (Ux0 × tiUx0) = A× tiD

if i ≥ 1. We identify A× tiD with A×D in a canonical way. By convention of §3.1,

o(A×D)(x,y) = −ds ds′ dx2 dy1.

38 TADAYUKI WATANABE

Let ν be a vector field on (A×D) \∆γ defined by

ν(x,y) = −(−s′, 0, x2, s′, y1, 0).This is perpendicular to ∆γ and points toward ∆γ . Thus the induced orientation

at the boundary of the blow-up is given by the formula

− ι(−s′, 0, x2, s′, y1, 0)(−ds ds′ dx2 dy1)= (ds+ ds′) ∧ (s′ dy1 ∧ dx2 − y1 ds′ ∧ dx2 + x2 ds

′ ∧ dy1).(4.4.1)

Let φ : (A×D) \∆Ux0→ R3 \ 0 be the map given by

φ(s, 0, x2, s′, y1, 0) = (s′, y1, 0)− (s, 0, x2) = (s′ − s, y1,−x2),

which induces the restriction of pr1 φ : N∆M\∆M → R3 \ 0. Then

φ∗ω(x, y) =1

‖φ(x, y)‖3(−(s′−s) dy1∧dx2+y1 (ds′−ds)∧dx2−x2 (ds′−ds)∧dy1

).

We consider the induced 2-form on the fiber F of N∆M\∆M . So we may impose

the relation s = −s′. Then the form φ∗ω induces

φ∗ω|F (x, y) =1

‖φ(x, y)‖3 (−2s′ dy1 ∧ dx2 + 2y1 ds

′ ∧ dx2 − 2x2 ds′ ∧ dy1).

Since o(∆γ) = ds′ + ds, the orientation of ST (γ) is given by

− 2

‖φ(x, y)‖3 (ds′ + ds) ∧ (s′ dy1 ∧ dx2 − y1 ds′ ∧ dx2 + x2 ds

′ ∧ dy1).

This is opposite to (4.4.1). Hence the smooth extensions of these forms to the

boundary give opposite orientations.

Case 2: ind γ = 2. Let x0, Ux0 , A, D be as in Case 2 of the proof of Lemma 3.3.

Then we have∫

R

X tY ∩ (Ux0 × tiUx0) = A× tiD

if i ≥ 1. By convention of §3.1,o(A ×D)(x,y) = ds ds′ dy1 dy2.

Thus the induced orientation at the boundary is given by

− ι(−s′, 0, 0, s′, y1, y2) ds ds′ dy1 dy2= (ds+ ds′) ∧ (s′ dy1 ∧ dy2 − y1 ds′ ∧ dy2 + y2 ds

′ ∧ dy1)(4.4.2)

Let φ : (A×D) \∆Ux0→ R3 \ 0 be the map given by

φ(s, 0, 0, s′, y1, y2) = (s′ − s, y1, y2),which induces the restriction of pr1 φ : N∆M

\∆M → R3 \ 0. Then

φ∗ω(x, y) =1

‖φ(x, y)‖3((s′− s) dy1 ∧dy2− y1 (ds′−ds)∧dy2 + y2 (ds′−ds)∧dy1

).

We consider the induced 2-form on the fiber F of N∆M. So we may impose the

relation s = −s′. Then the form φ∗ω induces

φ∗ω|F (x, y) =1

‖φ(x, y)‖3(2s′ dy1 ∧ dy2 − 2y1 ds

′ ∧ dy2 + 2y2 ds′ ∧ dy1

).

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 39

Since o(∆γ) = ds′ + ds, the orientation of ST (γ) is given by

2

‖φ(x, y)‖3 (ds′ + ds) ∧ (s′ dy1 ∧ dy2 − y1 ds′ ∧ dy2 + y2 ds

′ ∧ dy1).

This is equivalent to (4.4.2). Hence the smooth extensions of these forms to the

boundary give equivalent orientations.

The case where ind γ = 0 is the same as the case where ind γ = 2.

The reason that we must consider the irreducible factor γirr would be clear if one

considers a p-fold covering c : S1 → S1. A choice of base point v ∈ S1 gives a lift

cv : [0, 1]→ R of c in the universal covering π : R→ S1. The endpoints of cv defines

a point v0 ×Z v1 in R×Z R, which is defined analogously to M ×Z M . Conversely,

the set of points v0×Z v1 ∈ R×ZR such that π(v0) = π(v1) and v1 = tpv0(= v0−p)is of the form tp cv([0,

1p]), which can be written as tp cirr. This explains the reason

for the term tp(γ)ST (γirr).

Lemma 4.9. The face of ∂Q(ξ) at ST (M) contributes as s∗ξ(M).

Proof. It suffices to check that the induced orientation on the face of ∂Q(ξ) at

ST (M) is equivalent to the standard one on s∗ξ(M). So we consider a pair (x, y) ∈M×M of points both not close to any critical loci but close to each other. We choose

local coordinate systems (s, x1, x2) and (s′, y1, y2) around x and y respectively such

that ξx = ∂∂x1

and ξy = ∂∂y1

, the flow of ξ takes ∂∂xi

to ∂∂yi

and such that dκ( ∂∂s) > 0,

dκ( ∂∂s′

) > 0. By convention of §3.2, the orientation of Q(ξ) at (x, y) is given by

o(Q(ξ))(x,y) = (ds+ ds′) ∧ (dx1 + dy1) ∧ (dx2 + dy2) ∧ (−dy1)= (ds+ ds′) ∧ (dx2 + dy2) ∧ (−dx1) ∧ (−dy1).

The outward normal vector field on ∂Q(ξ) at b−1(∆M) is given by ξy − ξx =

∂∂y1− ∂

∂x1. Thus the induced orientation on the boundary is

ι

(∂

∂y1− ∂

∂x1

)(ds+ ds′) ∧ (dx2 + dy2) ∧ (−dx1) ∧ (−dy1)

= (ds+ ds′) ∧ (dx1 + dy1) ∧ (dx2 + dy2) = o(∆M)(x,x).

This is equivalent to the orientation of s∗ξ(M).

Proof of Theorem 4.7. By Lemma 4.5, the boundary ofQ(ξ) concentrates on ∂C2,Z(M).

The boundary of C2,Z(M) consists of faces made by the blow-up along b−1(∆M ).

By Lemma 4.9, the face made by the blow-up along b−1(∆M) contributes as s∗ξ(M).

The other faces correspond to the tangent sphere bundle over Z-cycles. Then

Lemma 4.8 gives the desired formula. That C(t)∆(M)Q(ξ) is a Λ-chain for a

product C(t) of cyclotomic polynomials follows from Lemma 4.2.

4.5. Relation with the Lefschetz zeta function. We shall see that the homol-

ogy class of the term∑

γ(−1)indγε(γ)tp(γ)ST (γirr) in the formula of Theorem 1.5

can be rewritten in terms of the Lefschetz zeta function.

40 TADAYUKI WATANABE

Proposition 4.10. Consider M as the mapping torus of an orientation preserving

diffeomorphism ϕ : Σ→ Σ, where Σ = κ−1(0). Then the following identity holds.

(4.5.1)∑

γ

(−1)indγε(γ) p(γirr) tp(γ) =tζ′ϕζϕ

,

where the sum is taken over equivalence classes of Z-cycles in M , or equivalently,

exp

(∑

γ

(−1)indγ ε(γ) p(γirr)

p(γ)tp(γ)

)= ζϕ.

Proof. By (1.2.2), the right hand side of (4.5.1) can be rewritten as

tζ′ϕζϕ

=

2∑

i=0

(−1)iTr tϕ∗i

1− tϕ∗i=

2∑

i=0

(−1)iTr tϕ♯i

1− tϕ♯i

=

2∑

i=0

(−1)i∞∑

k=1

tkTrϕk♯i.

Hence it suffices to check the identity

(4.5.2)∑

γind γ=i

ε(γ) p(γirr) tp(γ) =

∞∑

k=1

tkTrϕk♯i.

If i = 0 or 2, then ε(γ) = 1 for any Z-cycle γ and ϕ♯i is given by a permutation

matrix since a Z-cycle having a horizontal segment of index 0 or 2 cannot have

vertical segments. Thus the left hand side of (4.5.2) equals

∑

γ : irreducible

p(γ)(tp(γ) + t2p(γ) + · · · ) =∑

γ : irreducible

p(γ)tp(γ)

1− tp(γ)

for i = 0 or 2. On the other hand, if i = 0 or 2, ϕ♯i is represented by a direct

sum P1⊕P2⊕ · · · ⊕Pr of cyclic permutation matrices. Thus the right hand side of

(4.5.2) equalsr∑

j=1

∞∑

k=1

tkTrP kj =

r∑

j=1

mj

tmj

1− tmj,

where mj is the multiplicative order of Pj . This completes the proof of the identity

(4.5.2) for i = 0, 2.

The case i = 1 is more complicated since there may be Z-cycles that pass through

1/1-intersections. By definition of Θ(ξ), we have

∑

γind γ=1

ε(γ) p(γirr) tp(γ) =∞∑

k=1

tkTrΘ(ξ)k.

Indeed, the right hand side counts Z-cycles of period k with a base point on Σ. On

the other hand, the sum in the left hand side is over Z-cycles without base point.

The multiplicity of a Z-cycle γ without base point in the right hand side is exactly

p(γirr). This proves the identity above, which together with Lemma 4.1 completes

the proof.

Corollary 4.11. Let K be a knot in M such that 〈[dκ], [K]〉 = 1. Then

[∂Q(f)] = [sξ(M)] +tζ′ϕζϕ

[ST (K)]

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 41

in H3(∂C2,Z(M))⊗Λ Q(t).

Proof. This follows from Theorem 4.7, Proposition 4.10 and [ST (γirr)] = p(γirr)[ST (K)].

4.6. Equivariant linking number by counting Z-paths. Suppose that b1(M) =

1. Let α, β : S1 →M be two mutually disjoint embeddings that are nullhomologous

in M . Since the homotopy classes of α, β belong to [π1(M), π1(M)], there are lifts

α0, β0 : S1 → M of α, β respectively. According to Lescop’s result (Thorem 1.1),

the class of α0 ×Z β0 in H2(C2,Z(M)) ⊗Λ Q(t) is of the form n(α0, β0)[ST (∗)] forn(α0, β0) ∈ Q(t). The equivariant linking number ℓkZ(α, β) is defined as n(α0, β0).

(See [Les2] for detail.) This is well-defined modulo powers of t. If β0 bounds a

2-chain B0 in M , n(α0, β0) can be written explicitly as

n(α0, β0) = 〈α0, B0〉Z =∞∑

i=−∞

〈α0, t−iB0〉 ti,

where 〈·, ·〉 is the intersection pairing in M for piecewise smooth chains.

Let NZk (ξ;α0, β0) be the count of Z-paths in M from a point of Imα0 to a point

of tkImβ0, which is well-defined if α and β are generic.

Theorem 4.12. We have

ℓkZ(α, β).=

∞∑

k=−∞

NZk (ξ;α0, β0) t

k,

where.= means equal modulo powers of t.

Proof. As mentioned in [Les2], the coefficient n(α0, β0) of [ST (∗)] in the definition

of ℓkZ is given by the equivariant pairing

〈α0 ×Z β0, Q(ξ)〉Z =∑

k∈Z

〈tk(α0 ×Z β0), Q(ξ)〉 tk

=∑

k∈Z

〈α0 ×Z tkβ0, Q(ξ)〉 tk.

This gives the right hand side.

By Theorem 1.5 (2), there exists a Laurent polynomial A(t) ∈ Λ and a product

C(t) ∈ Λ of cyclotomic polynomials such that

ℓkZ(α, β).=

A(t)

C(t)∆(M).

Appendix A. Some facts on smooth manifolds with corners

We follow the convention in [BT, Appendix] for manifolds with corners, smooth

maps between them and their transversality. We write down some necessary terms

from [BT, Appendix], some of which are specialized than those in [BT, Appendix].

Definition A.1. (1) A map between manifolds with corners is smooth if it

has a local extension, at any point of the domain, to a smooth map from a

manifold without boundary, as usual.

42 TADAYUKI WATANABE

(2) Let Y, Z be smooth manifolds with corners, and let f : Y → Z be a bijective

smooth map. This map is a diffeomorphism if both f and f−1 are smooth.

(3) Let Y, Z be smooth manifolds with corners, and let f : Y → Z be a smooth

map. This map is strata preserving if the inverse image by f of a connected

component S of a stratum of Z is a union of connected components of strata

of Y .

(4) Let X,Y be smooth manifolds with corners and Z be a smooth manifold

without boundary. Let f : X → Z and g : Y → Z be smooth maps. Say

that f and g are (strata) transversal when the following is true: Let U and

V be connected components in stratums of X and Y respectively. Then

f : U → S and g : V → S are transversal.

The following proposition is a corollary of [BT, Proposition A.5].

Proposition A.2. Let X,Y be smooth manifolds with corners and Z be a smooth

manifold without boundary. Let f : X → Z and g : Y → Z be smooth maps that

are transversal. Then the fiber product

X ×Z Y = (x, y); f(x) = g(y) ⊂ X × Yis a smooth manifold with corners, whose strata have the form U×ZV where U ⊂ Xand V ⊂ Y are strata.

If f, g are inclusions then X×Z Y = (X×Y )∩∆Z = ∆X∩Y , which is canonically

diffeomorphic to X ∩ Y . Thus we obtain the following corollary.

Corollary A.3. Let X,Y be smooth manifolds with corners that are submanifolds

of a smooth manifold Z without boundary. Suppose that the inclusions X → Z and

Y → Z are transversal. Then the intersection X ∩ Y is a smooth manifold with

corners, whose strata have the form U ∩ V where U ⊂ X and V ⊂ Y are strata.

Proposition A.4. Let Z be a smooth manifold without boundary and let X be a

compact smooth submanifold of Z with corners. Suppose that dimX > 0. Then the

closure of the codimension 0 stratum IntX of X in Z agrees with X.

Proof. Let n = dimX and N = dimZ. Let

Rn〈m〉 = (x1, . . . , xn);x1 ≥ 0, . . . , xm ≥ 0 ⊂ Rn (m ≤ n).Choose an open cover Oλλ of X by small open N -disks Oλ in Z, say by open

ε-balls with respect to the geodesic distance for a Riemannian metric on Z for

small ε, so that for each λ there is a chart ϕλ : Oλ → ϕλ(Oλ) ⊂ RN such that the

restriction ϕλ|Oλ∩X : Oλ ∩X → RN factors as ι φλ where φλ : Oλ ∩X → Rn〈mλ〉is a chart and ι : Rn → RN is the inclusion (x1, . . . , xn) 7→ (x1, . . . , xn, 0, . . . , 0).

The codimension 0 stratum IntX of X is the union of preimages of ι(IntRn〈mλ〉)under charts ϕλ: IntX =

⋃λOλ ∩ ϕ−1

λ ι(IntRn〈mλ〉). The relation IntX ⊂ X fol-

lows immediately from definition of the closure and the compactness ofX . We prove

the converse. Since X is compact in Z, there is a finite subcover Oλ1 , . . . , Oλr of

X . Then we have

IntX =⋃r

i=1Oλi∩ ϕ−1

λiι(IntRn〈mλi

〉) = ⋃ri=1 ϕ

−1λiϕλi

(Oλi) ∩ ι(IntRn〈mλi

〉)⊃ ⋃r

i=1 ϕ−1λi

(ϕλi(Oλi

) ∩ ι(Rn〈mλi〉)) = ⋃r

i=1Oλi∩ ϕ−1

λiι(Rn〈mλi

〉) = X.

MORSE THEORY AND LESCOP’S EQUIVARIANT PROPAGATOR 43

Here at the first equality we have used the identity A1 ∪ · · · ∪ Ar = A1 ∪ · · · ∪ Ar

for arbitrary subsets A1, . . . , Ar (r < ∞) of a topological space, and between the

second and the third line we have used the relation O ∩ A ⊃ O ∩ A for O open, A

arbitrary, and the assumption n ≥ 1.

Appendix B. Blow-up

B.1. Blow-up of Ri along the origin. Let γ1(Ri) denote the total space of

the tautological oriented half-line ([0,∞)) bundle over the oriented Grassmannian

G1(Ri) ∼= Si−1. Namely,

γ1(Ri) = (x, y) ∈ Si−1 × Ri; ∃t ∈ [0,∞), y = tx.Then the tautological bundle is trivial and that γ1(Ri) is diffeomorphic to Si−1 ×[0,∞). We put

Bℓ0(Ri) = γ1(Ri)

and call Bℓ0(Ri) the blow-up of Ri along 0. Let π : γ1(Ri)→ Ri be the map defined

by π = pr2 ϕ in the following commutative diagram:

γ1(Ri)ϕ

//

π%%

Si−1 × Ri

pr2

pr1// Si−1

Ri

where ϕ : γ1(Ri)→ Si−1×Ri is the inclusion. We call π the projection of the blow-

up. Here, π−1(0) = ∂γ1(Ri) is the image of the zero section of the tautological

bundle pr1 ϕ : γ1(Ri)→ Si−1 and is diffeomorphic to Si−1.

Lemma B.1. (1) The restriction of π to the complement of π−1(0) = ∂γ1(Ri)

is a diffeomorphism onto Ri \ 0.(2) The restriction of ϕ to the complement of π−1(0) has the image in Si−1×Ri

whose closure agrees with the full image of ϕ from γ1(Ri).

B.2. Blow-up along a submanifold. When d > i ≥ 0, we put

BℓRi(Rd) = γ1(Ri)× Rd−i

(the blow-up of Rd along Ri) and define the projection : BℓRi(Rd) → Rd by

π × idRd−i . This can be straightforwardly extended to the blow-up BℓX(Y ) of

a manifold Y along a submanifold X by replacing the normal bundle with the

associated γ1(Rd)-bundle over X .

Acknowledgments.

This work was started during my stay at the Institut Fourier of the University

of Grenoble. I am grateful to the Institut for hospitality. Especially, I would like

to express my sincere gratitude to Professor Christine Lescop for kindly helping me

understand her work and for helpful conversations. I would like to thank Professor

Osamu Saeki for helpful comments and would like to thank the referee whose com-

ments were quite helpful for considerable improvement of this paper. I have been

supported by JSPS Grant-in-Aid for Young Scientists (B) 23740040 and by the

44 TADAYUKI WATANABE

JSPS International Training Program organized by Department of Mathematics,

Hokkaido University.

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Department of Mathematics, Shimane University, 1060 Nishikawatsu-cho, Matsue-shi,

Shimane 690-8504, Japan

E-mail address: tadayuki@riko.shimane-u.ac.jp