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MORSE THEORY AND LESCOP’S EQUIVARIANT
PROPAGATOR FOR 3-MANIFOLDS WITH b1 = 1 FIBERED
Abstract. For a 3-manifold M with b1(M) = 1 fibered over S1 and the fiberwise gradient ξ of a fiberwise Morse function on M , we introduce the notion of Z-path in M . A Z-path is a piecewise smooth path in M consisting of 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 function of M . The “moduli space” of Z-paths in M has a description like Chen’s iterated integrals and gives explicitly Lescop’s equivariant propagator, whose Poincaré dual generates the second cohomology of the configuration space of two points with rational function coefficients and which can be used to express Z-equivariant version of Chern–Simons perturbation theory for M by counting graphs. Counting Z-paths that connects two components in a nullhomologous link in M gives the equivariant linking number.
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 Poincaré–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.
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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 Marché’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 S 1, 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 C̃2(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
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
We will often write unions ⋃
s∈S Vs for continuous parameters s ∈ S, such as real numbers, as
Vs. When the parameter is at most countable, we will write
the unions as ∑
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
) ∈ 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∏
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):
dt log ζϕ(t) =
(−1)iTr ϕ∗i 1− tϕ∗i
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1.3. Equivariant configuration spaces. We recall some definitions from [Les2].
LetM be a closed or