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  • 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 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.

    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 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.

    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 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

    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)

    k tk

    ) ∈ 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

    dt log 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 or