Morse homotopy for the SU(2)–Chern–Simons perturbationtheory

TATSURO SHIMIZU

In this article, we give a generalized construction of the 2–loop term of the SU(2)–Chern–Simons perturbation theory by using the technique developed in [12]. Asan application, we give a Morse theoretic description of the 2–loop term of theSU(2)–Chern–Simons perturbation theory at a non-trivial connection.

1 Introduction

The G–Chern–Simons perturbation theory developed by M. Kontsevich in [5] andS. Axelrod and M. I. Singer in [1] gives a topological invariant of a closed oriented 3–manifold with a flat connection of the flat G-bundle over the given 3–manifold, where Gis an appropriate semi-simple Lie group. The Chern–Simons perturbation theory at thetrivial connection was well studied by G. Kuperberg and D. Thurston in [6], C. Lescopin [7], D. Moussard in [10] and C. Taubes in [13]. In particular, Kuperberg andThurston in [6] and Lescop in [7] showed that the Chern–Simons perturbation theoryat the trivial connection gives a universal finite type invariant of integral homology3–spheres. In 1999, R. Bott and A. S. Cattaneo gave a purely topological constructionof the Chern-Simons perturbation theory at a non-trivial connection in [2].

In 1996, K. Fukaya constructed the Morse homotopy invariant by using Morse functionsin [3]. The Morse homotopy invariant is an invariant of a closed oriented 3-manifoldwith two different flat connections of the trivial G-bundle on the given 3-manifold.Fukaya conjectured that the Morse homotopy invariant coincides with the 2–loop termof the Chern–Simons perturbation theory in some sense. In [4], M. Futaki pointed outthat Fukaya’s invariant depends on the choice of Morse functions by giving an explicitexample.

In 2012, T. Watanabe constructed the Morse homotopy invariant for the trivial con-nection in [15]. This is an invariant of rational homology 3-spheres. We showedthat Watanabe’s invariant coincides with the Chern–Simons perturbation theory at thetrivial connection in [12].

2 Tatsuro Shimizu

In this article, we give a generalized construction of the 2–loop term of the SU(2)–Chern–Simons perturbation theory at a non-trivial flat connection given by Bott andCattaneo in [2], by using the technique developed in [12]. The SU(2)–Chern–Simonsperturbation theory gives an invariant of the given 3–manifold with a flat connection.As an application of our generalized construction, we give a construction based onMorse functions of the 2–loop term of the SU(2)–Chern–Simons perturbation theory.

The invariant given by the Chern-Simons perturbation theory is the sum of the principalterm and the correction term. Both terms are invariants of a flat connection on a 3–manifold with an extra information on the 3-manifold. In the Bott and Cattaneoconstruction, they used a framing of the given 3–manifold as an extra infromation. Inthis article, we use three 3–cycles of the unit sphere bundle of the tangent bundle ofthe given 3-manifold. A framing gives three linearly independent unit vector fieldsand each vector field gives a 3-cycle of the unit sphere bundle of the tangent bundle ofthe 3-manifold. In this meaning, our construction is a generalization of the Bott andCattaneo construction.

Lescop defined an invariant of 3-manifolds with Betti number 1 in [8]. In the construc-tion of Lescop’s invariant, she used a similar technique as in the construction of theChern–Simons perturbation theory. Watanabe constructed an invariant of 3-manifoldwith Betti number 1 by using his Morse homotopy technique in [14]. It is expected thatthe Chern-Simons perturbation theory at a non-trivial connection is related to Lescop’sinvariant and Watanabe’s invariant in some sense.

The organization of this paper is as follows. In Section 2 we introduce some notations.In Section 3 we give a construction of the 2–loop term of the Bott and Cattaneo’s Chern–Simons perturbation theory. In Section 4 we discuss Chern–Simons perturbation theoryfor other Lie groups. In Section 5 we prove some lemmas and a theorem stated inSection 3. In Section 6 we give a Morse homotopy for the 2–loop term of the Chern–Simons perturbation theory as an application of the construction given in Section 3. InSection 7 we review compactifications of some moduli spaces used in Section 6. InSection 8 and Section 9 we prove Proposition 6.2 and Theorem 6.9 stated in Section 6.

Acknowledgments

The author expresses his appreciation to Professor Alberto Cattaneo for his kind andhelpful comments. The author would like to thank Professor Tadayuki Watanabe forhis helpful comments about the homology with local coefficients. The author alsoexpresses his appreciation to the referees for their valuable comments and suggestions

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 3

,in particular about the organization of this paper. This work was supported by JSPSKAKENHI Grant Number JP15K13437.

2 Homology with local coefficients

In this section, we prepare some notations about homology with local coefficients.

2.1 Chains with local coefficients

We first prepare the notation about the chains or cycles with local coefficients. LetX be a manifold and let Π(X) be the fundamental groupoid of X , namely Π(X) is acategory such that the objects of Π(X) are the elements of X and for any x, y ∈ X , amorphism from x to y is a path from x to y in X up to homotopy relative to x, y. Acovariant functor from Π(X) to a category of finite dimensional vector spaces is saidto be a local system on X . Let E be a local system of X . We denote by Ex the vectorspace corresponding to x ∈ X via the functor E . We denote by γ∗ : Ex

∼=→ Ey theisomorphism corresponding to a path γ ∈ Π(X)(x, y) from x ∈ X to y ∈ X via E .

For a finite dimensional representation ρ : π1(X) → GL+(Rn) of the fundamentalgroup. There is a unique local system Eρ such that for each x ∈ X , Ex ∼= Rn

and for γ ∈ Π(X)(x, x), γ∗ = ρ(γ). We call this local system Eρ the local systemcorresponding to ρ.

Let f : (A, a) → (X, f (a)) be a continuous map from a contractible compact orientedk–dimensional manifold A with a basepoint a ∈ A to X . Since A is contractible andoriented, the map f with e ∈ Ef (a) gives a k–chain in Ck(X; E) via an appropriatetriangulation of A. We denote by 〈f : A→ X, a; e〉 or 〈A, a; e〉 this chain.

More generally, a continuous map g : (B, b)→ (X, g(b)) from a compact k–dimensionalmanifold with a basepoint b ∈ B to X with e ∈ (Eg(b))g∗π1(B,b) gives a k–chain〈g : B → X, b; e〉 (or shortly 〈B, b; e〉). Here (Eg(b))g∗π1(B,b) is the invariant part ofEg(b) under the action of g∗π1(B, b) < π1(X, g(b)).

For a chain C = 〈f : A→ X, a; e〉, the support Supp(C) of C is defined by Supp(C) =

f (A).

Let π : X → M be a fiber bundle such that the typical fiber F is a compact orientedmanifold. Let c = 〈f : A→ M, a; e〉 ∈ Ck(M; E) be a k–chain of M . Then

π!c = 〈f : f ∗X → X, va; e〉 ∈ Ck+dim F(X;π∗E)

4 Tatsuro Shimizu

is a (k+dim F)–chain of X , where f : f ∗X → X is a bundle map induced by f : A→ Mand va ∈ π−1(a) is any point. Here f ∗X is the pull-back of π : X → M under f . Weorient f ∗X as a locally product of F and an open set of A. Here products are orientedby the order or the factors.

2.2 Intersection of chains

Let X be a compact n–dimensional manifold with boundary and let E be a local systemon X . Let c1, . . . , ck be singular chains of X such that:

(A) The degree of ci is ni : ci ∈ Cni(X; E) for each i,

(B) Supp(∂ci) ⊂ ∂X and there are open submanifolds U1, . . . ,Uk of ∂X satisfyingSupp(∂ci) ⊂ Ui and

⋂i Ui = ∅.

Here U is an open submanifold of ∂X if U is a (dim ∂X)–dimensional submanifoldand the boundary ∂U is a submanifold of ∂X , where U is a closure of U . Each ci givesa homology class [ci] ∈ Hni(X,Ui; E). We denote by [ci]P.D. ∈ Hn−ni(X, ∂X − Ui; E)the Poincare dual of [ci]. The cup product [c1]P.D. ∪ . . . ∪ [ck]P.D. is in

H∑

i(n−ni)(X,⋃

i

(∂X − Ui); E⊗k).

Thanks to the assumption (B),⋃

i(∂X − Ui) = ∂X . Then [c1]P.D. ∪ . . . ∪ [ck]P.D. ∈H

∑i(n−ni)(X, ∂X; E⊗k).

Definition 2.1 The intersection⋂

i ci of c1, . . . , ck is the homology class of degreen−

∑i(n− ni) of X given by⋂

i

ci = ([c1]P.D. ∪ . . . ∪ [ck]P.D.)P.D. ∈ Hn−∑

i(n−ni)(X; E⊗k).

More generally, the intersection⋂

i ci is also defined for ci with ∂ci 6⊂ ∂X . Letc1, . . . , ck be chains satisfying the following conditions:

(A) The degree of ci is ni : ci ∈ Cni(X; E) for any i,

(B) There are open submanifolds V1, . . . ,Vn ⊂ X and open submanifolds U1, . . . ,Un ⊂∂X such that Supp(∂ci) ⊂ Vi t Ui , Supp(ci) ∩ Vj = ∅ for any i 6= j and⋂

i Vi =⋂

i Ui = ∅.

In this case, each ci gives a homology class in Hni(X \ (⋃

i Vi),Ui ∪ ∂Vi; E). Then wedefine

⋂i ci = ([c1]P.D. ∪ . . . ∪ [ck]P.D.)P.D. ∈ Hn−

∑i(n−ni)(X \ (∪iVi); E⊗k).

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 5

Remark 2.2 Let ci ∈ Cni(X; E) be chains given as ci = 〈fi : Ai → X; ai, ei〉, whereai ∈ Ai, ei ∈ Efi(ai) for i = 1, . . . , k . We assume that f1, . . . , fk transversally intersectand fi|f−1

i (B) : f−1i (B) → X is injective for any i, where B = f1(A1) ∩ · · · ∩ fk(Ak).

In this case, the homology class⋂

i ci is represented by the geometric intersection ofc1, . . . , ck . Let B = B1 t . . .t Bl , where B1, . . . ,Bm are connected components of B.Take a basepoint bm ∈ Bm for each m = 1, . . . , l. Thus the homology class

⋂i ci is

represented as follows:

⋂i

ci =

l∑m=1

〈fj|f−1j (Bm) : f−1

j (Bm)→ X, f−1j (bm);⊗k

i=1(fi ◦ γmi )∗ei〉

for any j. Here γmi is a path from ai to f−1

i (bm) in Ai . Since ei ∈ (Efi(ai))(fi)∗π1(Ai,ai) ,

(fi ◦ γmi )∗ei ∈ Ebm is independent of the choice of the path γm

i . We remark that thehomology class of the right hand side of the equality is independent from j.

2.3 Conventions on orientations

Boundaries are oriented by the outward normal first convention. Products are orientedby the order of the factors. Interval [s, t] ⊂ R, for s < t , is oriented from s to t .

Let M be an oriented manifold of dimension n and let A and B be oriented submanifoldsof M of dimension k and l respectively. Let us assume that A and B intersecttransversally. Then A∩B is a k+l−n–dimensional submanifold of M . The orientationof A ∩ B determined from the above section is characterized by the following threeisomorphisms of oriented vector spaces:

TxA⊕ νxA ∼= TxM,

TxB⊕ νxB ∼= TxM and

Tx(A ∩ B)⊕ νxA⊕ νxB ∼= TxM,

where x ∈ A ∩ B is any point and νA is the normal bundle of A in M . The firstand second isomorphisms give orientations of the normal bundles and then the thirdisomorphism gives an orientation of Tx(A ∩ B).

Example 2.3 Let A = {(x1, x2, 0)},B = {(0, x2, x3)} ⊂ R3 . R3 is oriented by thestandard basis x1, x2, x3 . We assume that A is oriented by the basis x1, x2 of T0Aand B is oriented by the basis x2, x3 of T0B. Then the orientation of the intersectionA ∩ B = {(0, x2, 0)} is given by the basis x2 of T0(A ∩ B).

6 Tatsuro Shimizu

3 A construction of the SU(2)–Chern–Simons perturbationtheory

In this section we give a construction of the 2–loop term of the SU(2)–Chern–Simonsperturbation theory. This construction is a generalization of the construction given byBott and Cattaneo in [2] (See Remark 3.7 for the details).

Let M be an oriented closed 3–manifold and G be a semi-simple Lie group . AfterTheorem 3.5, we only consider the case of G = SU(2). Let ρ : π1(M) → G be arepresentation of π1(M). We consider a local system on M as a covariant functor fromthe fundamental groupoid of M to a category of finite dimensional vector spaces. Let Ebe the local system on M corresponding to the representation Ad◦ρ : π1(M)→ Aut(g),where g is the Lie algebra of G and Ad : G→ Aut(g) is the adjoint representation of G.We denote by Ex the object corresponding to x ∈ M , γ∗ the morphism correspondingto a path γ in M via the functor E . Ex is a vector space and γ∗ is an isomorphism.We assume that E is acyclic, namely Hk(M; E) = 0 for any k ∈ Z.

Example 3.1 Let ρ : π1M → SU(2) be a representation. Let Eρ be the local systemcorresponding to the representation Ad ◦ ρ. Then H1(M; Eρ) is isomorphic to theZariski tangent space of the representation variety Hom(π1M, SU(2))//SO(3). (Forexample, see [11, Lecture 15].) Therefore if ρ is isolated, then H1(M; Eρ) = 0. Thanksto Poincare duality, in this case, Eρ is acyclic. For example, any irreducible SU(2)representation ρ on the Seifert homology sphere Σ(2, 3, 5) is isolated.(See Corollary14.7 and Example after Corollary 14.7 in [11]). So Eρ is acyclic.

We denote by R the local system corresponding to the rank one trivial representation.E ⊗ E is a local system on M such that for any x ∈ M , (E ⊗ E)x = Ex ⊗ Ex . E isnaturally isomorphic to E∗x via the Killing form of g for any x ∈ M where E∗x is thedual space of Ex . Then Ex ⊗ Ex ∼= (E∗x ⊗ Ex)∗ for any x ∈ M . The evaluation map

ev : f ⊗ a 7→ f (a)

is in (E∗x ⊗ Ex)∗ = Hom(E∗x ⊗ Ex,R), where f ∈ E∗x , a ∈ Ex . Then we get a naturaltransformation map

c : R→ E ⊗ E

by c(1) = ev. In other words, c(1) =∑n

i=1 exi ⊗ ex

i . Here ex1, . . . , e

xn is an orthonormal

basis of Ex .

Let p1 : M2 → M and p2 : M2 → M be the projections. Let ∆ = {(x, x) | x ∈ M}be the diagonal. ∆ is oriented so that the diffeomorphism π1|∆ : ∆ → M (also

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 7

π2|∆ ) preserves the orientation. Let us denote by B`(M2,∆) the manifold with cornerobtained by the real blowing up of M2 along ∆. We denote by π : B`(M2,∆)→ M2

the blow down map. Let p1 : M2 → M and p2 : M2 → M be the projections. Thenp∗1E ⊗ p∗2E is a local system on M2 and F = π∗(p∗1E ⊗ p∗2E) is a local system onB`(M2,∆). We remark that p∗1E⊗ p∗2E|∆ = E⊗ E under the identification of M with∆ via the projection p1 . Similarly, F|π−1(∆) = π∗(E ⊗ E). Let T : B`(M2,∆) →B`(M2,∆) be an involution induced by the involution T0 : M2 → M2, (x, y) 7→ (y, x).We denote by H+

i (M2; p∗1E ⊗ p∗2E),H−i (M2; p∗1E ⊗ p∗2E) the +1 eigenspace and −1eigenspace of the induced map (T0)∗ : Hi(M2; p∗1E ⊗ p∗2E) → Hi(M2; p∗1E ⊗ p∗2E)respectively for any i. Similarly, we denote by H±i (B`(M2,∆); F) the ±1 eigenspaceof the induced map T∗; Hi(B`(M2,∆); F) → Hi(B`(M2,∆); F) respectively for anyi. We will use H±∗ (·),H+

±(·), C±∗ (·) and C∗±(·) in the same manner. We denoteby (E ⊗ E)+, (E ⊗ E)− the +1 eigenspace and −1 eigenspace of the involutionE ⊗ E → E ⊗ E, x ⊗ y 7→ y ⊗ x . For example, H+

1 (∆; E ⊗ E) = H1(∆; (E ⊗ E)+)because T0|∆ = id. However, H+

1 (π−1(∆);π∗(E ⊗ E)) 6= H1(π−1(∆);π∗(E ⊗ E)+).

We now define a propagator which plays an important role in the construction of theinvariant.

Definition 3.2 (propagator) A 4–cycle Σ ∈ C4(B`(M2,∆), ∂B`(M2,∆); F) is saidto be a propagator if there is a 3–cycle Σ∂

R ∈ C3(π−1(∆);R) such that the followingconditions hold:

(1) π∗[Σ∂R] = [∆] ∈ H3(∆;R),

(2) (T0)∗Σ∂R = Σ∂

R , T∗Σ = Σ and

(3) ∂Σ = c∗(Σ∂R).

Let Tr : E ⊗ E ⊗ E → R be the natural transformation defined by

Tr(a⊗ b⊗ c) = 〈[a, b], c〉

for a, b, c ∈ Ex , x ∈ M . Here 〈·, ·〉 is the Killing form of g and [·, ·] is the Lie bracketof g. Tr induces a natural transformation

Tr⊗2 : (p∗1E ⊗ p∗2E)⊗3 ∼= p∗1(E ⊗ E ⊗ E)⊗ p∗2(E ⊗ E ⊗ E)→ R⊗2 ∼= R

by Tr⊗2((ax ⊗ ay)⊗ (bx ⊗ by)⊗ (cx ⊗ cy)) = Tr(ax ⊗ bx ⊗ cx)⊗ Tr(ay ⊗ by ⊗ cy) forax, bx, cx ∈ Ex, ay, by, cy ∈ Ey , x ∈ M and y ∈ M . Tr⊗2 : (p∗1E ⊗ p∗2E)⊗3 → R alsoinduces Tr⊗2 : F⊗3 → R.

We take a triple of propagators (Σ1,Σ2,Σ3) satisfying the following conditions:

8 Tatsuro Shimizu

• There are submanifolds N1,N2 and N3 of π−1(∆) such that Ni ⊃ Supp(∂Σi)for i = 1, 2, 3,

• N1 ∩ N2 ∩ N3 = ∅.

We will call such a triple an admissible triple of propagators. Under these conditions,we can apply the intersection theory to Σ1,Σ2,Σ3 as in the usual homology theory.(More precisely, we consider the Poincare dual of each propagator. See Section 2.2)Σ1 ∩ Σ2 ∩ Σ3 is a 0-cycle in H0(B`(M2,∆); F⊗3). Then we have

Tr⊗2(Σ1 ∩ Σ2 ∩ Σ3) ∈ H0(B`(M2,∆);R) = R.

In fact this real number depends on the boundaries ∂Σ1, ∂Σ2 and ∂Σ3 . Then we nextintroduce a correction term.

Let STM be the unit sphere bundle of TM . Let FTM → STM be the tangent bundlealong the fiber of STM → M . We denote the Euler class of FTM → M by e(FTM) ∈H2(STM;R). Then e(FTM)P.D. ∈ H3(STM;R).

Let c1, c2, c3 ∈ C3(STM;R) be 3−cycles such that⋂

i Supp(ci) = ∅ and [c1] = [c2] =

[c3] = 12 e(FTM)P.D. . Let W0 be a compact oriented 4–manifold with ∂W0 = M . We

denote the Euler characteristic of W0 by χ(W0). We take non-negative integer k, lsatisfying 2l− k = χ(W0). Let W = W0]kCP2]lT4 . It is easy to check that χ(W) = 0and ∂W = M . Let TvW ⊂ TW be an oriented R3 –bundle satisfying TvW|M = TM .The total space of the unit sphere bundle STvW is an oriented 6-dimensional manifoldwith ∂STvW = STM . Let FTvW → STvW be the tangent bundle along the fiber ofSTvW → W and let e(FTvW) ∈ H2(STvW;R) be its Euler class. Then e(FTvW)P.D. ∈H4(STvW, ∂STvW;R). Take 4–cycles C1,C2,C3 ∈ C4(STvW, ∂STvW;R) such that∂Ci = ci and [Ci] = 1

2 e(FTvW)P.D. for i = 1, 2, 3. By the assumptions on c1, c2, c3 ,the intersection (C1 ∩ C2 ∩ C3) ∈ R = H0(STvW, ∂STvW;R) is well-defined.

Lemma 3.3 (Appendix of [12]) (C1 ∩ C2 ∩ C3) − 34 SignW ∈ R is independent of

the choice of C1,C2,C3 and W .

Proof Let W ′ ,C′1,C′2 and C′3 be another choice of W,C1,C2 and C3 respectively.

Then C1∪−C′1,C2∪−C′2 and C3∪−C′3 are 4-cycles of a closed 4-manifold W∪−W ′ .Ci ∪ −C′i represents the Euler class of FW∪−W′ for i = 1, 2, 3. Thanks to the casen = 1 of Proposition 5.3 in Appendix of [12],

(C1 ∩ C2 ∩ C3)− (C′1 ∩ C′2 ∩ C′3) = (C1 ∪ −C′1) ∩ (C2 ∪ −C′2) ∩ (C3 ∪ −C′3)

=68

Sign(W ∪ −W ′).

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 9

Then, Novikov additivity theorem implies that

(C1 ∩ C2 ∩ C3)− 34

SignW = (C′1 ∩ C′2 ∩ C′3)− 34

SignW ′.

Definition 3.4 I(c1, c2, c3) = (C1 ∩ C2 ∩ C3)− 34 SignW ∈ R.

There is a natural bundle isomorphism ψ : TM∼=→ (T(M2)|∆)/T∆ given by ψ(x, v) =

((x, x), (−v, v)), where x ∈ M and v ∈ TxM . ψ gives a bundle isomorphism betweenSTM and π−1(∆). We identify STM with π−1(∆) via this bundle isomorphism. Underthis identification, ∂Σ is a 3–cycle in C+

3 (STM;π∗(E⊗E)) for any propagator Σ. By thedefinition of propagator, there is a cycle Σ∂

R ∈ C+3 (STM;R) such that c∗(Σ∂

R) = ∂Σ.Since e(FTM)P.D. = 2 ∈ R = H+

3 (STM;R), we have [Σ∂R] = 1

2 e(FTM)P.D. . (SeeLemma 4.2 for the details).

From now on, we assume that G = SU(2).

Theorem 3.5 Let (Σ1,Σ2,Σ3) be an admissible triple of propagators. Let Σ∂R,i ∈

C3(STM;R) be the cycle such that c∗(Σ∂R,i) = ∂Σi for i = 1, 2, 3. Then

ZSU(2)(M,E; Σ1,Σ2,Σ3) = Tr⊗2(Σ1 ∩ Σ2 ∩ Σ3)− 6I(Σ∂R,1,Σ

∂R,2,Σ

∂R,3) ∈ R

is independent of the choice of (Σ1,Σ2,Σ3).

Theorem 3.5 will be proved in Section 5.

Definition 3.6 ZSU(2)(M,E) = ZSU(2)(M,E; Σ1,Σ2,Σ3).

Remark 3.7 In this article, we use three 3–cycles of the unit sphere bundle of the tan-gent bundle of the given 3-manifold. On the other hand, in [2], Bott and Cattaneo useda framing of the given 3–manifold to define the boundary conditions of propagators. Aframing gives three linearly independent unit vector fields and each vector field givesa 3-cycle of the unit sphere bundle of the tangent bundle of the 3-manifold. In thismeaning, our construction is a generalization of the Bott and Cattaneo construction.More precisely, the term Tr⊗2(Σ1 ∩ Σ2 ∩ Σ3) of ZSU(2)(M,E) equals to I(Θ,ρ,ρ)(M, f )the 2-loop term of Bott–Cattaneo’s Chern–Simons perturbative invariant (see [2, Ex-ample 2.2. and Theorem 2.5.]) when ∂Σ∂

R,i = p(f )−1(ai). Here p(f ) : STM → S2 is aprojection induced from a framing f and a1, a2, a3 ∈ S2 are arbitrary distinct points.

10 Tatsuro Shimizu

4 A sufficient condition for the existence of a propagator

In this section, we discuss the 2–loop term of the G–Chern–Simons perturbativeinvariants for more general semi-simple Lie groups G than SU(2). We define acohomology class o(E) such that there exists a propagator if and only if o(E) = 0.Finally, we show that there exists a propagator when G = SU(2).

Let G be a simply connected Lie group and let ρ : π1(M) → G be a representation.E is the local system corresponding to the representation Ad ◦ ρ : π1(M) → Aut(g).The local system F on B`(M2,∆), the natural transformation c : E ⊗ E → R andTr⊗2 : F⊗3 → R are defined as in the previous section.

There is a unique homology class

d(E) = ∂e−1∂−1c∗([∆]) ∈ H+3 (π−1(∆); F|π−1(∆)).

d(E) corresponds to c∗([∆]) ∈ H+3 (∆; E ⊗ E) under the following diagram, where

[∆] ∈ H3(∆;R) is the fundamental homology class.

H+4 (B`(M2,∆), ∂B`(M2∆); F) ∼=

e //

∂��

H+4 (M2,∆; p∗1E ⊗ p∗2E)

∂∼=��

H+3 (π−1(∆); F|π−1(∆)) // H+

3 (∆; E ⊗ E)

The left vertical line is a part of the long exact sequence of the pair (B`(M2,∆), ∂B`(M2,∆))and the right vertical line is a part of the long exact sequence of the pair (M2,∆). Thanksto the Kunneth formula, Hi(M2; p∗1E⊗ p∗2E) = 0 for any i. Then the right vertical lineis isomorphism. The top horizontal isomorphism e is the excision isomorphism.

Let s : ∆ → π−1(∆) be a section. Then [s(∆)] + T∗[s(∆)] is a homology class inH+

3 (π−1(∆);R).

Definition 4.1 o(E) = π∗(d(E) ∩ ([s(∆)] + T∗[s(∆)])) ∈ H1(∆; E ⊗ E).

Lemma 4.2 π∗ : H+3 (π−1(∆);R)→ H+

3 (∆;R) is an isomorphism.

Proof Fix a trivialization π−1(∆) ∼= ∆ × S2 . Since T|π−1(∆) : π−1(∆) → π−1(∆)reverses the orientation,

H+3 (π−1(∆);R) = H+

3 (∆× S2;R)∼= H−1 (∆;R)⊗ H−2 (S2;R)⊕ H+

3 (∆;R)⊗ H+0 (S2;R)

= H+3 (∆;R).

Obviously, π : π−1(∆)→ ∆ gives this isomorphism.

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 11

The following lemma says that o(E) is independent of the choice of s.

Lemma 4.3 The homology class [s(∆)] + T∗[s(∆)] is independent of the choice of asection s.

Proof [s(∆) ∪ Ts(∆)] is invariant under T . So, it is in H+3 (π−1(∆);R) and it

corresponds to 2 ∈ R under the isomorphism π∗ : H+3 (π−1(∆);R) ∼= R. Then the

homology class [s(∆) ∪ Ts(∆)] is independent of the choice of s.

Lemma 4.4 o(E) ∈ H−1 (∆; E ⊗ E).

Proof Since T|π−1(∆) : π−1(∆)→ π−1(∆) reverses the orientation,

H+3 (π−1(∆);R) ∼= H2

−(π−1(∆);R),

H+3 (π−1(∆);π∗(E ⊗ E)) ∼= H2

−(π−1(∆);π∗(E ⊗ E)),

H4+(π−1(∆);π∗(E ⊗ E)) ∼= H−1 (π−1(∆);π∗(E ⊗ E)).

d(E) is in H+3 (π−1(∆);π∗(E ⊗ E)) and [s(∆)] + T∗[s(∆)] is in H+

3 (π−1(∆);R).Then d(E) ∩ ([s(∆)] + T∗[s(∆)]) = (d(E)P.D. ∪ ([s(∆)]P.D. + T∗[s(∆)]P.D.)P.D. ∈H−1 (π−1(∆);π∗(E ⊗ E)). Since the following diagram commutes, the push forwardπ∗ : H1(π−1(∆);π∗(E⊗E))→ H1(∆; E⊗E) induces the map π∗ : H−1 (π−1(∆);π∗(E⊗E))→ H−1 (∆; E ⊗ E).

π−1(∆)T|π−1(∆)//

π|π−1(∆)

��

π−1(∆)

π|π−1(∆)

��∆

T0|∆ // ∆

Therefore o(E) = π∗(d(E) ∩ ([s(∆)] + T∗[s(∆)])) ∈ H−1 (∆; E ⊗ E).

Lemma 4.5 If o(E) = 0, there exist propagators.

Proof Let s : ∆ → π−1(∆) be a section. The cycle 12 ([s(∆)] + T∗[s(∆)]) ∈

C+3 (π−1(∆);R) satisfies the condition (1) of Definition 3.2. Set Σ∂

R = 12 ([s(∆)] +

T∗[s(∆)]). Let V = ker(π∗ : H+3 (π−1(∆);π∗(E ⊗ E)) → H+

3 (∆; E ⊗ E)). By thedefinition of d(E),

d(E)− c∗[Σ∂R] ∈ V.

We fix a trivialization π−1(∆) ∼= ∆× S2 . Under the trivialization,

H+3 (π−1(∆);π∗(E⊗E)) ∼= H+

3 (∆; E⊗E)⊗H+0 (S2;R)⊕H−1 (∆; E⊗E)⊗H−2 (S2;R).

12 Tatsuro Shimizu

Then we have

V ∼= H−1 (∆; E ⊗ E)⊗ H2(S2;R)ϕ∼= H−1 (∆; E ⊗ E).

For any a ∈ H−1 (∆; E ⊗ E), π∗((a ⊗ [S2]) ∩ Σ∂R) = a. Then the last isomorphism ϕ

is given by ϕ(·) = π∗(· ∩ Σ∂R).

We next show that π∗(c∗[Σ∂R] ∩ [Σ∂

R]) = 0. Since the homology class 2[Σ∂R] =

[s(∆)] + T∗[s(∆)] is independent from the chosen section s, it is enough to check thatπ∗(c∗[Σ∂

R]∩ [Σ∂R]) = 0 for a section s. Take p 6= ±q ∈ S2 . Let 1

2 (∆×{−p, p}) = Σ∂R

for the first [Σ∂R] and let 1

2 (∆× {−q, q}) = Σ∂R for the second [Σ∂

R]. The supports ofthese cycles are disjoint. Then c∗([Σ∂

R])∩[Σ∂R] = 1

4 c∗([∆×{±p}])∩[∆×{±q}] = 0.

Therefore

0 =12

o(E)− π∗(c∗[Σ∂R] ∩ [Σ∂

R])

= π∗(d(E) ∩ [Σ∂R])− π∗(c∗[Σ∂

R] ∩ [Σ∂R])

= π∗((d(E)− c∗[Σ∂R]) ∩ [Σ∂

R])

= ϕ(d(E)− c∗[Σ∂R]).

Thus we have d(E) = c∗[Σ∂R]. Since d(E) is in the image of the map

H4(B`(M2,∆), ∂B`(M2,∆); F)→ H3(∂B`(M2,∆); F),

d(E) = c∗[Σ∂R] implies that

c∗[Σ∂R] ∈ Im(H4(B`(M2,∆), ∂B`(M2,∆); F)→ H3(∂B`(M2,∆); F)).

So we can take Σ0 ∈ C4(B`(M2,∆); F) satisfying ∂Σ0 = c∗Σ∂R . Then Σ = 1

2 (Σ0 +

T(Σ0)) ∈ C+4 (B`(M2,∆), ∂B`(M2,∆); F) is a propagator.

Lemma 4.6 If G = SU(2), then H−1 (∆; E ⊗ E) = 0.

Proof Since T0|∆ = id, T0 acts only on the local coefficient Ex ⊗ Ex for any (x, x) ∈∆. Then H−1 (∆; E ⊗ E) = H1(∆; (E ⊗ E)−). The Lie bracket [·, ·] of su2 induces anatural transformation map b : E ⊗ E → E, b(x ⊗ y) = [x, y]. For each (x, x) ∈ ∆,b : Ex ⊗ Ex → Ex is surjective because su2 is semi-simple. Then dim(kerb) = 6.Since b(T(x⊗ y)) = −b(x⊗ y), (Ex ⊗ Ex)+ is a subspace of kerb. On the other handdim((Ex⊗Ex)+) = 6. Therefore (E⊗E)− ∼= E . Then H1(∆; (E⊗E)−) = H1(∆; E) =

0.

Thanks to Lemma 4.4 and Lemma 4.6, the following proposition holds.

Proposition 4.7 If G = SU(2), then o(E) = 0.

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 13

5 Well-definedness of ZSU(2)(M,E)

Lemma 5.1 Let (Σ1,Σ2,Σ3) and (Σ′1,Σ′2,Σ

′3) be admissible triples of propagators

such that ∂Σi = ∂Σ′i and [Σi − Σ′i] = 0 ∈ H+4 (B`(M2,∆); F) for any i = 1, 2, 3.

Then Tr⊗2(Σ1 ∩ Σ2 ∩ Σ3) = Tr⊗2(Σ′1 ∩ Σ′2 ∩ Σ′3) ∈ R ∼= H0(B`(M2,∆),R).

Proof It is enough to show that Tr⊗2(Σ1∩Σ2∩Σ3−Σ′1∩Σ2∩Σ3) = 0. The intersec-tion of chains Σ2 ∩Σ3 gives a cycle [Σ2 ∩Σ3] in H2(B`(M2,∆), ∂B`(M2,∆); F⊗ F)and the cycle [Σ1−Σ′1] is in H4(B`(M2,∆); F). Then the homology class representedby

(Σ1 ∩ Σ2 ∩ Σ3)− (Σ′1 ∩ Σ2 ∩ Σ3) = (Σ1 − Σ′1) ∩ (Σ2 ∩ Σ3)

is given as the intersection of homology classes:

[(Σ1 − Σ′1) ∩ (Σ2 ∩ Σ3)] = [Σ1 − Σ′1] ∩ [Σ2 ∩ Σ3].

The homology class [Σ1 − Σ′1] ∩ [Σ2 ∩ Σ3] is in H0(B`(M2,∆); F⊗3). Thanks to theassumption that [Σ1 − Σ′1] = 0, we have

[Σ1 − Σ′1] ∩ [Σ2 ∩ Σ3] = 0 ∩ [Σ2 ∩ Σ3] = 0.

ThereforeTr⊗2((Σ1 ∩ Σ2 ∩ Σ3)− (Σ′1 ∩ Σ2 ∩ Σ3)) = 0.

Lemma 5.2 If G = SU(2), then H+4 (B`(M2,∆); F) = 0.

Proof By the Poincare duality and the excision isomorphism, H+4 (B`(M2,∆); F) ∼=

H2−(B`(M2,∆), ∂B`(M2,∆); F) ∼= H2

−(M2,∆; p∗1E ⊗ p∗2E). Since H∗(M; E) = 0, theKunneth formula shows that H∗(M2; p∗1E⊗p∗2E) = 0. The cohomology exact sequenceof the pair (M2,∆) implies that H2

−(M2,∆; p∗1E⊗p∗2E) ∼= H1−(∆; E⊗E). By the same

reason as in the proof of Lemma 4.6, H1−(∆; E ⊗ E) = H1(∆; (E ⊗ E)−) = 0.

Corollary 5.3 If G = SU(2), ZSU(2)(M,E; Σ1,Σ2,Σ3) ∈ R is independent of thechoice of an admissible triple of propagators (Σ1,Σ2,Σ3).

Lemma 5.4 Tr⊗2(c(1)⊗ c(1)⊗ c(1)) = 6.

14 Tatsuro Shimizu

Proof Let e1, e2, e3 be an orthonormal basis of su(2) with respect to the Killing form.

For example, e1 = 12

(0 ii 0

), e2 = 1

2

(0 −11 0

), e3 = 1

2

(i 00 −i

)∈ su(2).

Tr⊗2(c(1)⊗ c(1)⊗ c(1))

= Tr⊗2(∑

i,j,k∈{1,2,3}

(ei ⊗ ei)⊗ (ej ⊗ ej)⊗ (ek ⊗ ek))

= Tr⊗2(∑

i,j,k∈{1,2,3}

〈[ei, ej], ek〉〈[ei, ej], ek〉)

= Tr⊗2(∑

{i,j,k}={1,2,3}

|〈[ei, ej], ek〉|2

= 6|〈[e1, e2], e3〉|2

= 6.

Proof of Theorem 3.5 Let Σ?i be an alternative choice of Σi . Let Σ?,∂

R,i ∈ C3(STM;R)

be the cycle satisfying c∗(Σ?,∂R,i ) = ∂Σ?

i for i = 1, 2, 3. Let C?i ,Ci be 4-cycles in

C4(STvW;R) satisfying ∂C?i = Σ?,∂R,i , ∂Ci = Σ∂

R,i for i = 1, 2, 3. Let [−1, 0] ×STM ⊂ STvW, {0} × STM = ∂STvX be a collar of STM in STvW . We identifySTvW \ ([−1, 0]× STM) with STvW by stretching the collar. Since [Σ?,∂

R,i ] = [Σ∂R,i](=

12 e(FTM)P.D) (see Theorem 3.5 below), there exists a 4-chain Σ∂

R,i ∈ C4([−1, 0] ×

STM;R) such that ∂Σ∂R,i = Σ?,∂

R,i−Σ∂R,i ∈ C3({0}×STMt−{−1}×STM;R). Thanks

to Lemma 3.3, without loss of generality, we may assume that C?i |STvW\([−1,0]×STM) =

Ci and C?i |[−1,0]×STM = Σ∂R,i . Then

I(Σ?,∂R,1,Σ

?,∂R,2,Σ

?,∂R,3)− I(Σ∂

R,1,Σ∂R,2,Σ

∂R,3)

= C?1 ∩ C?2 ∩ C?3 ∩ ([−1, 0]× STM).

Let [−1, 0] × π−1(∆) ⊂ B`(M2,∆), {0} × π−1(∆) = ∂B`(M2,∆) be a collar ofπ−1(∆) in B`(M2,∆). Thanks to Lemma 5.1, without loss of generality, we mayassume that

Σ?i |B`(M2,∆)\([−1,0]×π−1(∆)) = Σi and Σ?

i |[−1,0]×π−1(∆) = C?i |[−1,0]×STM . Then

Tr⊗2(Σ?1,Σ

?2,Σ

?3)− Tr⊗2(Σ1,Σ2,Σ3)

= Tr⊗2(Σ?1 ∩ Σ?

2 ∩ Σ?3 ∩ ([−1, 0]× π−1(∆))

= Tr⊗2(c(1)⊗ c(1)⊗ c(1))(C?1 ∩ C?2 ∩ C?3 ∩ ([−1, 0]× STM))

= 6(C?1 ∩ C?2 ∩ C?3 ∩ ([−1, 0]× STM)).

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 15

Therefore Tr⊗2(Σ1 ∩ Σ2 ∩ Σ3) − 6I(Σ∂R,1,Σ

∂R,2,Σ

∂R,3) = Tr⊗2(Σ?

1 ∩ Σ?2 ∩ Σ?

3) −6I(Σ?,∂

R,1,Σ?,∂R,2,Σ

?,∂R,3).

6 Morse homotopy for the SU(2)–Chern–Simons perturba-tion theory

In [12] we introduced the Morse theoretic description (Morse homotopy) of a propaga-tor for the Chern–Simons perturbation theory at the trivial connection. This descriptionis deeply inspired by [3] and [15] and related to [9]. In this section, we give a Morsetheoretic description of ZSU(2)(M,E).

Let f : M → R be a Morse function. We denote by Critj(f ) the set of critical pointsof index j of f and set Crit(f ) = tj=0,1,2,3Critj(f ). We denote by ind(p) the index ofp ∈ Crit(f ).

Let (Φtf : M

∼=→ M)t∈R be the one–parameter family of diffeomorphisms associated togradf . Let Ap and Dp be the ascending manifold of p and descending manifold of prespectively for p ∈ Crit(f ):

Ap = {x ∈ M | limt→−∞

Φtf (x) = p},

Dp = {x ∈ M | limt→∞

Φtf (x) = p}.

We orient Ap and Dp by imposing the condition TpAp ⊕ TpDp = TpM for any p. LetM(p, q) be the set of all trajectories connecting p ∈ Critk(f ) and q ∈ Critk+1(f ):

M(p, q) = {γ : R→ M | dγ/dt = gradf , limt→−∞

γ(t) = p, limt→∞

γ(t) = q}/R.

Here r ∈ R acts as r ·γ(t) = γ(t + r). We consider each γ ∈M(p, q) as a path from pto q. We give a sign ε(γ) ∈ {+,−} for each γ ∈M(p, q) as follows: The orientationof γ is given as a component of the intersection γ ⊂ Ap ∩ Dq . If the orientation isfrom q to p, we set ε(γ) = +. If the orientation is from p to q, we set ε(γ) = −.

Let

(C∗(f ; E), ∂) = · · · 0 ∂4→ C3(f ; E) ∂3→ C2(f ; E) ∂2→ C1(f ; E) ∂1→ C0(f ; E) ∂0→ 0

be the Morse-Smale complex of the Morse function f with the acyclic local system E ,namely Cj(f ; E) = ⊕q∈Critj(f )Eq and

∂j(x) =∑

p∈Critj−1(f )

∑γp,q∈M(p,q)

ε(γ)(γp,q)−1∗ x

for any x ∈ Eq, q ∈ Critj(f ).

16 Tatsuro Shimizu

Definition 6.1 ([3],[15]) A family of homomorphisms g = (gk : Ck(f ; E) →Ck+1(f ; E))k∈Z is said to be a combinatorial propagator if

∂k+1 ◦ gk + (−1)kgk−1 ◦ ∂k = idCk(f ;E)

for all k ∈ Z.

A combinatorial propagator g gives a morphism gp,q : Ep → Eq for p ∈ Critk(f ), q ∈Critk+1(f ) by gp,q = πEq ◦ gk|Ep , where πEq : ⊕r∈Critk+1(f )Er → Eq is the projection.

Let M2(f ) be the set of all pairs of points in M connected by a trajectory in M :

M2(f ) = {(x,Φtf (x)) | t ≥ 0, x ∈ M} ⊂ M2.

We orient M2(f ) so that the inclusion −[0, 1] × M → M2(f ), (t, x) 7→ (x,Φtf (x))

preserves the orientation. It is known that there is a compactification M2(f ) whichconsists of all broken (and not broken) trajectories and M2(f ) gives a 4-chain

MB`2 (f ) ∈ C4(B`(M2; ∆); F)

of B`(M2; ∆). In Section 7, we will discuss the details of the construction of MB`2 (f ).

(See also [9, Lemma 4.3] or [15, Proposition 3.4]).

Similarly, there are compactifications Ap of Ap and Dq of Dq . Ap and Dq areobtained by adding all the broken trajectories. For p ∈ Critk(f ) and q ∈ Critk+1(f ),(gp,q ⊗ 1) is a morphism from Ep ⊗ Ep to Eq ⊗ Ep . Therefore

〈Dq ×Ap, (q, p); (gp,q ⊗ 1)c(1)〉

is a 4-chain in C4(M2, p∗1E ⊗ p∗2E). We denote by

〈(Dq ×Ap)B`, (q, p); (gp,q ⊗ 1)c(1)〉 ∈ C4(B`(M2,∆); F)

the 4–chain given from the closure of π−1(Dq×Ap \∆) in B`(M2,∆). See Section 7and [15, Proposition 3.14] for more details. Let

M0(f ) = MB`2 (f ) +

∑p,q∈Crit(f ),ind(p)=ind(q)−1

〈(Dq ×Ap)B`, (q, p); (gp,q ⊗ 1)c(1)〉.

Proposition 6.2 M0(f ) is a 4–cycle of C4(B`(M2,∆), ∂B`(M2,∆); F).

This proposition will be proved in Section 8. Let

M(f ) =12

(M0(f ) + T∗(M0(f ))) ∈ C+4 (B`(M2,∆); F).

For v ∈ TxM \ {0}, we denote by [v] ∈ STxM the image of v under the projectionTxM \ {0} → STM .

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 17

Set

∆(f ) = {[gradyf ] ∈ STyM | y ∈ M \ Crit(f )}

t{−[gradyf ] ∈ STyM | y ∈ M \ Crit(f )} ⊂ STM

Here {[gradyf ] ∈ STyM | y ∈ M \ Crit(f )} and {−[gradyf ] ∈ STyM | y ∈ M \ Crit(f )}are the closure of {[gradyf ] ∈ STyM | y ∈ M \ Crit(f )} and {−[gradyf ] ∈ STyM | y ∈M \ Crit(f )} in STM respectively. It is known that ∆(f ) is a 3–cycle of C3(STM;R)(See [12, Lemma 4.5]). Therefore

c∗(∆(f )) = 〈∆(f ), (x, [gradx f ]); c(1)〉

is a 3-cycle of C+3 (STM;π∗(E ⊗ E)), where x ∈ M \ Crit(f ) is any point.

Set

z0(f ) =∑

p,q∈Crit(f );ind(p)=ind(q)−1

〈Dq ∩ Ap, q; ((gp,q ◦∑

γ∈M(p,q)

γ−1∗ )⊗ id)c(1)〉,

z(f ) =12

(z0(f )− T∗z0(f )) ∈ C−1 (M; E ⊗ E).

By using these cycles, the boundary ∂M(f ) ∈ C3(π−1(∆);π∗(E⊗E)) = C3(STM;π∗(E⊗E)) can be described as follows:

Lemma 6.3∂M(f ) =

12〈∆(f ), (x, [gradx f ]); c(1)〉+ π!z(f ).

In particular, z(f ) is a 1-cycle of C−1 (M; E ⊗ E).

This lemma will be proved in Section 8.

We next introduce the linking number of 1–cycles of C−1 (M; E⊗E)(= C−1 (∆; E⊗E)).Let c1, c2 ∈ C−1 (M,E ⊗ E) be 1-cycles such that Supp(c1) ∩ Supp(c2) = ∅. SinceH−1 (M,E ⊗ E) = 0, there is a 2–chain D1 ∈ C−2 (M; E ⊗ E) such that ∂D1 = c1 .Then we have a 0–cycle D1 ∩ c2 ∈ C0(M; (E ⊗ E)⊗2). Let R : (E ⊗ E)⊗2 →R,R((x1 ⊗ x2)⊗ (y1 ⊗ y2)) = 〈[x1, y1], [x2, y2]〉 = Tr⊗2((x1 ⊗ x2)⊗ (y1 ⊗ y2)⊗ c(1)).

Lemma 6.4 R∗(D1 ∩ c2) ∈ R = H0(M;R) is independent of the choice of D1 .

Proof Let D′1 be the alternative choice of D1 . D1−D′1 is a 2–cycle of D−2 (M; E⊗E).Since H−2 (M; E⊗E) = 0, D1∩c2−D′1∩c2 = [D1−D′1]∩[c2] = 0 ∈ H−0 (∆; (E⊗E)⊗2).Therefore R∗(D1 ∩ c2)− R∗(D′1 ∩ c2) = R∗(0) = 0.

18 Tatsuro Shimizu

Definition 6.5 lk(M,E)(c1, c2) = R∗(D1 ∩ c2).

Lemma 6.6 lk(M,E)(c1, c2) = lk(M,E)(c2, c1).

Proof Take 2-chains D1,D2 ∈ C−2 (M; E ⊗ E) satisfying ∂D1 = c1 and ∂D2 = c2 .Since the boundary of any 1-chain of H1(M;R) represents 0 in H0(M;R), we have

0 = ∂R∗(D1 ∩ D2)

= R∗(∂(D1 ∩ D2))

= R∗(D1 ∩ ∂D2)− R∗(∂D1 ∩ D2)

= R∗(D1 ∩ ∂D2)− R∗(D2 ∩ ∂D1)

= lk(M,E)(c1, c2)− lk(M,E)(c2, c1).

Definition 6.7 A triple of Morse functions (f1, f2, f3) is said to be generic when thefollowing conditions hold:

• Supp(z(fi)) ∩ Supp(z(fj)) = ∅ for any i 6= j and

• ∆(f1) ∩∆(f2) ∩∆(f3) = ∅.

Remark 6.8 A generic triple of Morse functions in the product space C∞(M,R)3 ofa map space C∞(M,R) is generic in the above sense.

Theorem 6.9 For any generic triple of Morse functions (f1, f2, f3),

ZSU(2)(M,E) = Tr⊗2(M(f1) ∩M(f2) ∩M(f3))− 34

I(∆(f1),∆(f2),∆(f3))

−∑

i<j∈{1,2,3}

lk(M,E)(z(fi), z(fj)).

7 Compactifications of M2(f ), Ap and Dq

In this section we review the compactifications of M2(f ), Ap and Dq . See [9, Lemma4.3] and [15, Proposition 3.14] for more details of construction and properties of thesecompactifications.

Set

Mbr2 (f ) = tp1,p2,··· ,pk∈Crit(f )(Dp1 ×M(p1, p2)× · · · ×M(pk−1, pk)×Apk ).

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 19

Here ind(pi) < ind(pi+1) for i = 1, . . . , k − 1. An element of Mbr2 (f ) is called

broken trajectory. We consider each broken trajectory (x, γ1, · · · , γk−1, y) ∈ Dp1 ×M(p1, p2)× · · · ×M(pk−1, pk)×Apk as a piecewise smooth path from x to y in M .

Let π1, π2 : M2(f ) tMbr2 (f )→ M be the map defined as

π1(γ) = x, π2(γ) = y

for a (broken) trajectory γ form x to y.

Let γ = (x, γ1, · · · , γk−1, y) ∈ Dp1×M(p1, p2)×· · ·×M(pk−1, pk)×Apk be a brokentrajectory. For sufficiently small neighborhoods B(x),B(y) of x, y in M , we can definea map

ϕ : (π1 × π2)−1(B(x)× B(y))→ B(x)× f−1(a1)× · · · × f−1(ak−1)× B(y)

byϕ(γ) = (π1(γ), γ ∩ f−1(a1), · · · , γ ∩ f−1(ak−1), π2(γ)),

where ai = (f (pi) + f (pi+1))/2 for i = 1, · · · , k − 1 and γ is a (broken) trajectorybelonging to (π1 × π2)−1(B(x)× B(y)).

Let M2(f ) = M2(f ) tMbr(f ) as a set. We define a neighborhood of γ in M2(f ) as

(π1 × π2)−1(B(ϕ(γ)))

for a small neighborhood B(ϕ(γ))of ϕ(γ) in B(x)× f−1(a1)× · · · × f−1(ak−1)×B(y).

Let J : M2(f )× [0, 1]→ M2(f ) be a map defined by

J(γ, t) = γ ∩ f−1([t(f (π1(γ))− f (π2(γ))) + f (π2(γ)), f (π2(γ))]).

J is a homotopy from a retract r : M2(f ) → ∆, r(γ) = (π2(γ), π2(γ)) to the identitymap. Therefore we can define a 4-chain of M2

〈M2(f ), (x, x); c(1)〉= 〈π1 × π2|M2(f ) : M2(f )→ M2, (x, x); c(1)〉 ∈ C4(M2; p∗1E ⊗ p∗2E).

Here (x, x) is any point in ∆. The restriction map π1 × π2|(π1×π2)−1(M2\∆) : M2(f ) ∩(π1 × π2)−1(M2 \∆) → M2 \∆ can be extended to the closure B`(M2; ∆) and thengives a chain of C4(B`(M2; ∆); F). We will denote by

MB`2 (f ) ∈ C4(B`(M2,∆); F)

this chain.

We next review the compactification of Ap,Dq . For any p, q ∈ Crit(f ), {p}×Ap andDq × {q} are subspaces of Mbr

2 (f ). We denote by {p} × Ap,Dq × {q} the closure of

20 Tatsuro Shimizu

{p} × Ap,Dq × {q} in Mbr2 (f ) respectively. Since the map J′ : {p} × Ap × [0, 1] →

{p} × Ap given by

J′(γ, t) = (p, γ ∩ f−1([f (π1(γ)), t(f (π2(γ))− f (π1(γ))) + f (π1(γ))]))

is a homotopy from the retract r′ : {p} × Ap× [0, 1]→ {p}×{p} to the identity map,r′ is a deformation retract. Then {p} × Ap and also Dq × {q} are contractible. Thenthe continuous map

π2|{p}×Ap: {p} × Ap → M

andπ1|Dq×{q} : Dp × {q} → M

with any ep ∈ Ep(= Eπ2(p,p)) and eq ∈ Eq(= Eπ1(q,q)) give chains

〈{p} × Ap, (p, p); ep〉 ∈ C3−ind(p)(M; E),

〈Dq × {q}, (q, q); eq〉 ∈ Cind(q)(M; E).

For simplicity, we denote

〈Ap, p; ep〉 = 〈{p} × Ap, (p, p); ep〉,

〈Dq, q; eq〉 = 〈Dq × {q}, (q, q); eq〉.

For any eq,p ∈ Eq ⊗ Ep , we have a (3− ind(p) + ind(q))-chain

〈Dq ×Ap, (q, p); eq,p〉

in C3−ind(p)+ind(q)(M2; p∗1E⊗p∗2E). Since Dq is transversally intersect to Ap for q 6= p,there is a (3− ind(p) + ind(q))-chain

〈(Dq ×Ap)B`, (q, p); eq,p〉 ∈ C3−ind(p)+ind(q)(B`(M2; ∆); F)

obtained from Dq×Ap when q 6= p. In particular when ind(p) = ind(q)− 1, we havea 4-chain

〈(Dq ×Ap)B`, (q, p); (gp,q ⊗ 1)c(1)〉 ∈ C4(B`(M2; ∆); F).

8 Boundaries of the compactifications

We first discuss a stratification of the compactification M2(f ) of M2(f ).

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 21

Lemma 8.1 M2(f ) is a compact manifold with a stratification such that its codimensionk stratum ∂kM2(f ) is given by

∂kM2(f ) = tp1,p2,...,pk∈Crit(f )(Dp1 ×M(p1, p2)× · · · ×M(pk−1, pk)×Apk ).

In particular∂M2(f ) = ∂1M2(f ) = −

∑p∈Crit(f )

Dp ×Ap

as an oriented manifold.

Proof Let (x, y) ∈ Dp ×Ap ⊂ Mbr2 (f ) for p ∈ Crit(f ). Let Lx ⊂ M be a sufficiently

small n − ind(p) dimensional disk embedded in M which transversally intersects toDp at x = Lx ∩ Dp . Similarly, let Ly ⊂ M be a sufficiently small ind(p) dimensionaldisk embedded in M which transversally intersects to Ap at y = Ly ∩ Ap . By usingthe local model of the Morse singularity, we can check that the manifold

π−11 (Lx) ∩ π−1

2 (Ly) ⊂ M2(f )

is diffeomorphic to [0, 1) and 0 ∈ [0, 1) corresponds to the broken trajectory connectingx and y. Thus a small neighborhood U of (0, 0, 0) in TxDp × TyAp × [0, 1) gives alocal coordinate of M2(f ) around (x, y). Under the local coordinate, (a neighborhoodof (x, y) in) Dp × Ap corresponds to U ∩ (TxDp × TyAp × {0}). This implies thatDp × Ap is a part of the boundary of M2(f ). We can check that the orientation ofDp ×Ap as the product of Dp and Ap is opposite to the orientation as a boundary ofM2(f ). Since Dp ×Ap = Dp ×Ap , we have

∂M2(f ) = ∂1M2(f ) = −∑

p∈Crit(f )

Dp ×Ap.

Similarly, Dp1 ×M(p1, p2) ×Ap2 is a part of the boundaries of both Dp1 ×Ap1 andDp2 ×Ap2 . Thus Dp1 ×M(p1, p2)×Ap2 is a part of ∂2M2(f )

We can check that ∂kM2(f ) = tp1,p2,...,pk∈Crit(f )(Dp1×M(p1, p2)×· · ·×M(pk−1, pk)×Apk ) by iteration.

Lemma 8.2

∂〈M2(f ), (x, x); c(1)〉 \∆ = −∑

p∈Crit(f )

〈Dp ×Ap, (p, p); c(1)〉 \∆.

Proof This lemma is a direct consequence of Lemma 8.1

22 Tatsuro Shimizu

Lemma 8.3 For any eq ∈ Eq, ep ∈ Ep ,

∂〈Dq, q; eq〉 =∑

p′∈Critind(q)−1(f )

∑γp′,q∈M(p′,q)

ε(γp′,q)〈Dp′ , p′; (γp′,q)−1∗ eq〉,

∂〈Ap, p; ep〉 =∑

q′∈Critind(p)+1(f )

∑γp,q′∈M(p,q′)

(−1)ind(p)+1ε(γp,q′)〈Aq′ , q′; (γp,q′)∗ep〉.

Proof We show the first assertion. Thanks to Lemma 8.1,

Dq × {q} =⋃

p′∈Crit(f );ind(p′)<ind(q)

(Dp′ ×M(p′, q)× {q}

)⊂ M2

as a manifold (we do not care about the orientations yet). Thus we have

∂〈Dq × {q}, (q, q), eq〉

=⋃

p′∈Crit(f );ind(p′)<ind(q)

〈π1 : Dp′ ×M(p′, q)× {q} → M, (p′, γp′,q, q); (γp′,q)−1∗ eq〉.

Here, γp′,q ∈M(p′, q) is any trajectory.

When ind(p′) < ind(q) − 1, dimM(p′, q) = ind(q) − ind(p′) − 1 ≥ 1. Thus themap π1 : Dp′ ×M(p′, q)× {q} → M factors through a less than (dimDq − 1)-dimensional manifold Dp′ × {q}. Then the chain 〈π1 : Dp′ ×M(p′, q)× {q} →M, (p′, γ, q); γ−1

∗ eq〉 can be removed from the chain ∂Dq by using an appropriatetriangulation of Dq .

Therefore we have

∂〈Dq × {q}, (q, q); eq〉

=⋃

p′∈Crit(f );ind(p′)=ind(q)−1

〈π1 : Dp′ ×M(p′, q)× {q} → M, (p′, γp′,q, q); (γp′,q)−1∗ eq〉.

Since all broken trajectories connecting p′ and q are inM(p′, q) (namely, they are notbroken) for any p′ ∈ Crit(f ) satisfying ind(p′) = ind(q)− 1,

Dp′ ×M(p′, q)× {q}

=⋃

r1,...,rk∈Crit(f )

Drk ×M(rk, rk−1)× . . .×M(r1, p′)×M(p′, q)× {q}

= Dp′ ×M(p′, q)× {q}= tγ∈M(p′,q)Dp′ × {q}.

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 23

By our orientation conventions, the orientation of Dp′ × γ × {q} as a part of theboundary of Dq coincides with the orientation of ε(γ)Dp′ . Therefore

∂〈Dq, q; eq〉 =∑

p′∈Critind(q)−1(f )

∑γp′,q∈M(p′,q)

ε(γp′,q)〈Dp′ , p′; (γp′,q)−1∗ eq〉.

Thanks to Lemma 8.2 and Lemma 8.3, we have the following corollary:

Corollary 8.4 When ind(p) = ind(q)− 1,

∂〈Dq ×Ap, (q, p); (gp,q ⊗ 1)c(1)〉

=∑

q′∈Critind(q)(f )

∑γp,q′∈M(p,q′)

(−1)ind(q)ε(γp,q′)〈Dq ×Aq′ , (q, q′); (gp,q ⊗ (γp,q′)∗)c(1)〉

+∑

p′∈Critind(p)(f )

∑γp′,q∈M(p′,q)

ε(γp′,q)〈Dp′ ×Ap, (p′, p); (((γp′,q)−1∗ ◦ gp,q)⊗ 1)c(1)〉.

8.1 Proof of Proposition 6.2

〈M2(f ), (x, x); c(1)〉 +∑

p,q∈Crit(f );ind(p)=ind(q)−1〈Dq × Ap, (q, p); (gp,q ⊗ 1)c(1)〉 is a4–chain of M2 . It is sufficient to show that

Supp

∂〈M2(f ), (x, x); c(1)〉+

∑p,q∈Crit(f );ind(p)=ind(q)−1

〈Dq ×Ap, (q, p); (gp,q ⊗ 1)c(1)〉

⊂ ∆.

24 Tatsuro Shimizu

By Corollary 8.4,

∂〈Dq ×Ap, (q, p); (gp,q ⊗ 1)c(1)〉

=∑

q′∈Critind(q)(f )

∑γp,q′∈M(p,q′)

(−1)ind(q)ε(γp,q′)〈Dq ×Aq′ , (q, q′); (gp,q ⊗ (γp,q′)∗)c(1)〉

+∑

p′∈Critind(p)(f )

∑γp′,q∈M(p′,q)

ε(γp′,q)〈Dp′ ×Ap, (p′, p); (((γp′,q)−1∗ ◦ gp,q)⊗ 1)c(1)〉

=∑

q′∈Critind(q)(f )

〈Dq ×Aq′ , (q, q′); (−1)ind(q)ε(γp,q′)((gp,q ◦∑

γp,q′∈M(p,q′)

(γp,q′)−1∗ )⊗ 1)c(1)〉

+∑

p′∈Critind(p)(f )

〈Dp′ ×Ap, (p′, p); ε(γp′,q)((∑

γp′,q∈M(p′,q)

(γp′,q)−1∗ ◦ gp,q)⊗ 1)c(1)〉

=∑

q′∈Critind(q)(f )

〈Dq ×Aq′ , (q, q′); (((−1)ind(q)gp,q ◦ ∂q′,p)⊗ 1)c(1)〉

+∑

p′∈Critind(p)(f )

〈Dp′ ×Ap, (p′, p); ((∂q,p′ ◦ gp,q)⊗ 1)c(1)〉.

Therefore

∂

∑p,q∈Crit(f );ind(p)=ind(q)−1

〈Dq ×Ap, (q, p); (gp,q ⊗ 1)c(1)〉

=

∑k

∑r,r′∈Critk(f )

〈Dr ×Ar′ , (r, r′);

(∑

q′∈Critk+1(f )

∂q′,r ◦ gr′,q′ + (−1)k∑

p′∈Critk−1(f )

gp′,r ◦ ∂r′,p′)⊗ 1

c(1)〉

=∑

k

∑r,r′∈Critk(f )

〈Dr ×Ar′ , (r, r′);(πEr ◦ (∂k+1 ◦ gk + (−1)kgk−1 ◦ ∂k)⊗ 1

)|Er′ c(1)〉

=∑

k

∑r,r′∈Critk(f )

〈Dr ×Ar′ , (r, r′); δr,r′c(1)〉

=∑

r∈Crit(f )

〈Dr ×Ar, (r, r); c(1)〉

Here, πEr : ⊕s∈Critk(f )Es → Er is the projection.

On the other hand, thanks to Lemma 8.2,

∂〈M2(f ), (x, x); c(1)〉 \∆ = −∑

p∈Crit(f )

〈Dp ×Ap, (p, p); c(1)〉 \∆.

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 25

Therefore

Supp

∂〈M2(f ), (x, x); c(1)〉+

∑p,q∈Crit(f );ind(p)=ind(q)−1

〈Dq ×Ap, (q, p); (gp,q ⊗ 1)c(1)〉

⊂ ∆.

8.2 Proof of Lemma 6.3

Since limt→0(x,Φtf (x)) = (x, [gradxf ]) in B`(M2,∆) for x 6∈ Crit(f ), ∂MB`

2 (f ) =

{(x, [gradxf ]) | x 6∈ Crit(f )}.

Thanks to the Morse-Smale condition, each 4-manifold Dq×Ap transversally intersectsto ∆ along Dq ∩ Ap , for q, p ∈ Crit(f ) satisfying ind(p) = ind(q) − 1. By ourorientation conventions,

∂〈(Dq×Ap)B`, (q, p); (gp,q⊗1)c(1)〉 = π!∑

γ∈M(p,q)

〈Dq∩Ap, q; ((gp,q ◦γ−1∗ )⊗1)c(1)〉.

Finally, we have

∂M(f ) =12〈∆(f ), (x, [gradxf ]); c(1)〉+ π!z(f ).

9 Proof of Theorem 6.9

Since H−1 (∆; E ⊗ E) = 0, there are 2–chains D(f1),D(f2),D(f3) ∈ C−2 (∆; E ⊗ E)satisfying ∂D(fi) = z(fi) for i = 1, 2, 3. Let [−1, 0] × ∂B`(M2,∆) be a collar of∂B`(M2,∆) ⊂ B`(M2,∆) with {0}× ∂B`(M2,∆) = ∂B`(M2,∆). For −t ∈ [−1, 0],we denote by

π[−t,0] : [−t, 0]× ∂B`(M2,∆)→ ∂B`(M2,∆),

π−t : {−t} × ∂B`(M2,∆)→ ∂B`(M2,∆)

the projections. We note that π◦π[−t,0] = (π|∂B`(M2,∆))◦π[−t,0] : [−t, 0]×∂B`(M2,∆)→∆ is the projection. We identify F|[−1,0]×∂B`(M2,∆) with π∗[−1,0]π

∗(E ⊗ E). We mayassume that

M(fi) ∩ ([−1, 0]× ∂B`(M2,∆)) = [−1, 0]× ∂M(fi)

( = π![−1,0]∂M(fi)).

for any i = 1, 2, 3. We set

• D′(f1) = π!−1/4π

!D(f1)− π![−1/4,0]π

!z(f1) ∈ C4(B`(M2,∆), π−1(∆); F),

26 Tatsuro Shimizu

• D′(f2) = π!2/4π

!D(f2)− π![−2/4,0]π

!z(f2)) ∈ C4(B`(M2,∆), π−1(∆); F),

• D′(f3) = π!3/4π

!D(f3)− π![−3/4,0]π

!z(f3)) ∈ C4(B`(M2,∆), π−1(∆); F),

• M(f1)′ =M(f1)− D′(f1),

• M(f2)′ =M(f2)− D′(f2) and

• M(f3)′ =M(f3)− D′(f3).

Then the triple (M(f1)′,M(f2)′,M(f3)′) is a triple of admissible propagators with∂M(fi)′ = 1

2〈∆(fi), (x, [gradxf ]); c(1)〉 = c∗( 12∆(fi)). Thus we have

ZSU(2)(M,E) = Tr⊗2(M(f1)′ ∩M(f2)′ ∩M(f3)′)− 6I(12

∆(f1),12

∆(f2),12

∆(f3))

= Tr⊗2(M(f1)′ ∩M(f2)′ ∩M(f3)′)− 34

I(∆(f1),∆(f2),∆(f3)).

Lemma 9.1 (1) Tr⊗2(D′(f1)∩M(f2)∩M(f3)) = lk(M,E)(z(f1), z(f2))+lk(M,E)(z(f1), z(f3)),

(2) Tr⊗2(M(f1) ∩ D′(f2) ∩M(f3)) = lk(M,E)(z(f1), z(f2)) + lk(M,E)(z(f2), z(f3)),

(3) Tr⊗2(M(f1) ∩M(f2) ∩ D′(f3)) = lk(M,E)(z(f1), z(f3)) + lk(M,E)(z(f2), z(f3)),

(4) Tr⊗2(D′(f1) ∩ D′(f2) ∩M(f3)) = lk(M,E)(z(f1), z(f2)),

(5) Tr⊗2(D′(f1) ∩M(f2) ∩ D′(f3)) = lk(M,E)(z(f1), z(f3)),

(6) Tr⊗2(M(f1) ∩ D′(f2) ∩ D′(f3)) = lk(M,E)(z(f2), z(f3)) and

(7) Tr⊗2(D′(f1) ∩ D′(f2) ∩ D′(f3)) = 0.

The following lemma is used in the proof of Lemma 9.1.

Lemma 9.2 For any a+, b+, c+ ∈ (E ⊗ E)+ , a−, b−, c− ∈ (E ⊗ E)− and anyσ1, σ2, σ3 ∈ {+,−} ∼= Z/2 with σ1σ2σ3 = −, Tr⊗2(aσ1 ⊗ bσ2 ⊗ cσ3) = 0.

Proof Since σ1σ2σ3 = −,

T⊗30 (aσ1 ⊗ bσ2 ⊗ cσ3) = σ1aσ1 ⊗ σ2bσ2 ⊗ σ3cσ3 = −aσ1 ⊗ bσ2 ⊗ cσ3 .

On the other hand, Tr⊗2 ◦ T⊗30 = Tr⊗2 . Then Tr⊗2(aσ1 ⊗ bσ2 ⊗ cσ3) = 0.

Proof of Lemma 9.1 We first prove (1). Because of the assumptions about M(f2)and M(f3),

Tr⊗2(D′(f1) ∩M(f2) ∩M(f3))

= Tr⊗2(D′(f1) ∩ π![−1,0]∂M(f2) ∩ π!

[−1,0]∂M(f3))

= Tr⊗2(D′(f1) ∩ (12π!

[−1,0]c∗(∆(f2)) + π![−1,0]π

!z(f2)) ∩ (12π!

[−1,0]c∗(∆(f3)) + π![−1,0]π

!z(f3))).

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 27

By the construction,

c∗(∆(f2)), c∗(∆(f3)) ∈ C3(π−1(∆);π∗((E ⊗ E)+))

and

z(f2), z(f3),D′(f1) ∈ C2(∆; (E ⊗ E)−).

Thus

D′(f1) ∩ 12π!

[−1,0]c∗(∆(f2)) ∩ 12π!

[−1,0]c∗(∆(f3))

∈ C0([−1, 0]× ∂B`(M2,∆);π∗((E ⊗ E)− ⊗ (E ⊗ E)+ ⊗ (E ⊗ E)+)),

D′(f1) ∩ π![−1,0]π

!z(f2) ∩ π![−1,0]π

!z(f3)

∈ C0([−1, 0]× ∂B`(M2,∆);π∗((E ⊗ E)− ⊗ (E ⊗ E)− ⊗ (E ⊗ E)−)).

Thanks to Lemma 9.2, we have

Tr⊗2(D′(f1) ∩ 12π!

[−1,0]c∗(∆(f2)) ∩ 12π!

[−1,0]c∗(∆(f3))) = 0,

Tr⊗2(D′(f1) ∩ π![−1,0]π

!z(f2) ∩ π![−1,0]π

!z(f3)) = 0.

Therefore,

Tr⊗2(

D′(f1) ∩(

12π!

[−1,0]c∗(∆(f2)) + π![−1,0]π

!z(f2))∩(

12π!

[−1,0]c∗(∆(f3)) + π![−1,0]π

!z(f3)))

= Tr⊗2(D′(f1) ∩ π![−1,0]c∗(

12

∆(f2)) ∩ π![−1,0]π

!z(f3))

+ Tr⊗2(D′(f1) ∩ π![−1,0]π

!z(f2) ∩ π![−1,0]c∗(

12

∆(f3)))

= Tr⊗2(π!−1/4π

!D(f1) ∩ π![−1,0]c∗(

12

∆(f2)) ∩ π![−1,0]π

!z(f3))

− Tr⊗2(π![−1/4,0]π

!z(f1) ∩ π![−1,0]c∗(

12

∆(f2)) ∩ π![−1,0]π

!z(f3))

+ Tr⊗2(π!−1/4π

!D(f1) ∩ π![−1,0]π

!z(f2) ∩ π![−1,0]c∗(

12

∆(f3)))

− Tr⊗2(π![−1/4,0]π

!z(f1) ∩ π![−1,0]π

!z(f2) ∩ π![−1,0]c∗(

12

∆(f3)))

= Tr⊗2(π!−1/4(π!D(f1) ∩ c∗(

12

∆(f2)) ∩ π!z(f3))) + Tr⊗2(π![−1/4,0](π

!z(f1) ∩ c∗(12

∆(f2)) ∩ π!z(f3)))

+ Tr⊗2(π!−1/4(π!D(f1) ∩ π!z(f2) ∩ c∗(

12

∆(f3)))) + Tr⊗2(π![−1/4,0](π

!z(f1) ∩ π!z(f2) ∩ c∗(12

∆(f3))))

28 Tatsuro Shimizu

For generic f1, f2, f3 , Supp(z(f1))∩ Supp(z(f2)) = ∅ and Supp(z(f1))∩ Supp(z(f3)) = ∅.Thus

Tr⊗2(π!−1/4(π!D(f1) ∩ c∗(

12

∆(f2)) ∩ π!z(f3))) + Tr⊗2(π![−1/4,0](π

!z(f1) ∩ c∗(12

∆(f2)) ∩ π!z(f3)))

+ Tr⊗2(π!−1/4(π!D(f1) ∩ π!z(f2) ∩ c∗(

12

∆(f3)))) + Tr⊗2(π![−1/4,0](π

!z(f1) ∩ π!z(f2) ∩ c∗(12

∆(f3))))

= Tr⊗2(π!−1/4(π!D(f1) ∩ c∗(

12

∆(f2)) ∩ π!z(f3))) + Tr⊗2(π!−1/4(π!D(f1) ∩ π!z(f2) ∩ c∗(

12

∆(f3)))).

By the way,

Tr⊗2(π!−1/4(π!D(f1) ∩ π!z(f2) ∩ c∗(

12

∆(f3)))

= Tr⊗2((D(f1) ∩ z(f2))⊗ c(1))

= lk(M,E)(z(f1), z(f2)).

So we finally obtain

Tr⊗2(D′(f1) ∩M(f2) ∩M(f3)) = lk(M,E)(z(f1), z(f2)) + lk(M,E)(z(f1), z(f3)).

We next prove (4). For generic f1, f2 , Supp(z(f1)) ∩ Supp(z(f2)) = ∅. Therefore

Tr⊗2(D′(f1) ∩ D′(f2) ∩M(f3)) = Tr⊗2(D′(f1) ∩ D′(f2) ∩ π![−1,0]∂M(f3))

= Tr⊗2(π!1/4π

!D(f1) ∩ π!1/4π

!z(f2) ∩ π!1/4∂M(f3))

= Tr⊗2(π!1/4π

!D(f1) ∩ π!1/4π

!z(f2) ∩ 12π!

1/4c∗(∆(f3)))

+ Tr⊗2(π!1/4π

!D(f1) ∩ π!1/4π

!z(f2) ∩ π!1/4π

!z(f3)).

Since, for generic f2, f3 , Supp(z(f2)) ∩ Supp(z(f3)) = ∅, the second term of the lastequation is zero. Then

Tr⊗2(D′(f1) ∩ D′(f2) ∩M(f3))) = lk(M,E)(z(f1), z(f2)).

(7) is followed from the fact that Supp(D′(f1)) ∩ Supp(D′(f2)) ∩ Supp(D′(f3)) = ∅ forgeneric f1, f2, f3 .

Morse homotopy for the SU(2)–Chern–Simons perturbation theory 29

Proof of Theorem 6.9 Thanks to Lemma 9.1,

Tr⊗2(M(f1)′ ∩M(f2)′ ∩M(f3)′)

= Tr⊗2((M(f1)− D′(f1)) ∩ (M(f2)− D′(f2)) ∩ (M(f3)− D′(f3)))

= Tr⊗2(M(f1) ∩M(f2) ∩M(f3))

−Tr⊗2(D′(f1) ∩M(f2) ∩M(f3))

−Tr⊗2(M(f1) ∩ D′(f2) ∩M(f3))

−Tr⊗2(M(f1) ∩M(f2) ∩ D′(f3))

+Tr⊗2(D′(f1) ∩ D′(f2) ∩M(f3))

+Tr⊗2(D′(f1)′ ∩M(f2) ∩ D′(f3))

+Tr⊗2(M(f1) ∩ D′(f2) ∩ D′(f3))

−Tr⊗2(D′(f1) ∩ D′(f2) ∩ D′(f3))

= Tr⊗2(M(f1) ∩M(f2) ∩M(f3))−∑i<j

lk(M,E)(z(fi), z(fj)).

Since ZSU(2)(M,E) = Tr⊗2(M(f1)′ ∩M(f2)′ ∩M(f3)′)− 34 I(∆(f1),∆(f2),∆(f3)), we

conclude the proof of Theorem 6.9.

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30 Tatsuro Shimizu

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Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 JAPAN

shimizu@kurims.kyoto-u.ac.jp