KONTSEVICH's CHARACTERISTIC CLASSES Contents Introduction 1

KONTSEVICH’S CHARACTERISTIC CLASSES

ARAMINTA AMABEL

Contents Introduction 1 Acknowledgments 2 1. Classifying Space 2 2. Graph Complex 4 3. Compactiﬁcations of Conﬁgurations Spaces 4 4. Propagators 6 5. Kontsevich’s Characteristic Classes 7 References 12

Introduction The goal of this note is to construct characteristic classes for framed sphere bundles. We follow the overview of these classes in [3] and [4]. The main results of [3] and [4] show that many of these characteristic classes are nontrivial. Watanabe constructs framed odd-dimensional sphere bundles over spheres for which Kontsevich’s characteristic classes from graphs with vertices of valence 3 do not vanish. The characteristic classes are defined roughly as follows, with terms to be defined later. Let π : E → B be a framed sphere bundle, say with fiber M a homology d-sphere, with d odd. We will define a characteristic class for π given the data of an oriented graph Γ with n vertices and ∗ m edges whose vertices have valence at least 3. We want to define a closed form in ΩdR(B). Consider the bundle πn : ECn(π) → B associated to π whose fiber is the configuration space of n points in M, Cn(M). We would like to say that integration along the fibers gives us a map from the deRham complex of ECn(π) to the deRham complex of B, but the fibers of π2 are not compact. To remedy this, we instead consider the bundle ECn(π) → B with fiber the Fulton- MacPherson-Kontsevich compactification of the configuration space. Each edge e of Γ determines a map φe : ECn(π) → EC2(π) which, on the fibers, picks out the two points of the configuration corresponding to the labels of the vertices of the edge e. Wedging together all of the maps, we ∗ ∗ obtain a map ΩdR(ECn(π)) → ΩdR(EC2(π)). In §4, we will define a form ω on ECn(π) called a “propagator.” The deRham form on B of π corresponding to Γ is then the image of ω under the composite V ∗ R d−1 e φe m(d−1) fiber m(d−1)−nd ΩdR (EC2(π)) −−−−→ ΩdR (ECn(π)) −−−→ ΩdR (B). In §1 we describe the classifying space for such sphere bundles; i.e., the space whose cohomology contains the classes we will construct. We will construct characteristic classes for each oriented graph of valence at least 3. Such graphs fit together into a chain complex which we describe in §2. In §3, we describe a compactification of configuration spaces originally defined by Fulton-MacPherson and Kontsevich [1], [2]. In §4 we define propagators on families of configuration spaces. Finally, in 1 §5, we define the characteristic classes and prove Kontsevich’s theorem (Thm. 5.1) showing that the classes are closed, do not depend on the choice of propagator, and are natural with respect to bundle maps of framed bundles. Acknowledgments. These notes were written for a course on Diffeomorphisms of Disks taught at Harvard in the fall of 2017 by Alexander Kupers. Many mistakes, misunderstandings, and typos in earlier drafts of these notes were found by Kupers.

1. Classifying Space We begin by giving precise definitions of the type of bundles we want to study. Characteristic classes of such bundles should live in the cohomology of the classifying space of such bundles. We end this section by giving a model for such a classifying space. Definition 1. A homology d-sphere is a smooth closed d-dimensional manifold M so that ∼ d Hi(M; R) = Hi(S ; R) for all i. Let M be a homology d-sphere. Fix a point ∞ ∈ M. Then M \ ∞ is acyclic and hence paralleliz- d able. Fix U∞ ⊂ M a small disk around ∞ ∈ M and also fix a diffeomorphism h∞ : U∞ → V∞ ⊂ R ∪ {∞} of balls centered at ∞, sending ∞ to ∞. ∼ = d Definition 2. Say a trivialization κ: T (M \∞) −→ R ×(M \∞) is standard near ∞ if over U∞ \∞, d the trivialization κ is the pullback along h∞ of the standard framing ι on T R . Equivalently, if the diagram dh∞ d T (M \ ∞)|U∞\∞ / (T R )|V∞\∞

κ ι d d R × (U∞ \ ∞) / R × (V∞ \ ∞) Id×h∞ commutes.

Fix such a trivialization κ that is standard near ∞. Let Diﬀ(M,U∞) denote the group of diﬀeomorphisms of M that are the identity on U∞.

Definition 3. An (M,U∞)-bundle on a smooth manifold B is a smooth fiber bundle π : E → B with fiber M and structure group Diff(M,U∞). ∼ Remark. Let DM = M \ U∞. Then Diff(M,U∞) = Diff(DM , ∂DM ). We will alternate between the two descriptions. Note that Watanabe uses (M,U∞) in [3] and (DM , ∂) in [4].

A(M,U∞)-bundle π : E → B has a section at infinity, s: B → E. Let U ∞ denote a tubular neighborhood of s(B) in E. Note that Watanabe continues to denote this tubular neighborhood by U∞. Recall that if B is a manifold, the vertical tangent bundle T fib(E) of a bundle π : E → B is the kernel of dπ : TE → TB. fib Definition 4. Say a trivialization τ of (T E)|U∞\s(B) is induced from κ if the following diagram commutes fib τ d d (T E)| / U ∞ \ s(B) × R × B / (M \ ∞) × R × B U ∞\s(B) ∼=

∼= κ×1 π ) U ∞ \ s(B) // (U∞ \ ∞) × B / T (M \ ∞) × B ∼= 0-section 2 ﬁb Deﬁnition 5. A vertical framing of a (M,U∞)-bundle π : E → B is a trivialization τ of (T E)|E\s(B) of the vertical tangent bundle outside the image of s, such that τ is induced by κ on U ∞ \ s(B). Let Fr(M) be the space of framings on T (M \ ∞) that agree with κ near ∞.

Proposition 1.1. Let B be a manifold, and M,U∞ as above. Homotopy classes of maps

B → EDiﬀ(M,U∞) ×Diﬀ(M,U∞) Fr(M) are in bijection with equivalence classes of vertically framed (M,U∞)-bundles.

In other words, B^Diff(M,U∞) := EDiff(M,U∞) ×Diff(M,U∞) Fr(M) is a model for the classifying space of vertically framed (M,U∞)-bundles. We will prove Proposition as a special case of the following lemma. Lemma 1.2. Let G be a topological group and X a G-space. For a manifold B, homotopy classes of maps [B,EG ×G X] are in bijection with concordance classes of triples (P, π, f) where π : P → B is a principle G-bundle and f : P → X is a G-equivariant map. Proof. Given such a triple (P, π, f), consider the classifying map ψ : B → BG for π,

ψˆ P / EG

π ψ B / BG deﬁne a map σ : B → EG ×G X by ˆ σ(π(x)) = ψ(x) ×G f(x)(1)

This is well-defined since π is surjective and ψˆ and f are G-equivariant. Conversely, given a map σ : B → EG ×G X, define π : P → B to be the pullback of EG → BG along ψ := πX ◦ σ where πX : EG ×G X → BG collapses X. Define f again by the formula 1.

Proof of Proposition 1. Take G = Diﬀ(M,U∞) and X = Fr(M). By Lemma 1.2, homotopy classes of maps

B → EDiff(M,U∞) ×Diff(M,U∞) Fr(M) are in bijective correspondence with concordance classes of triples (P, π, f) where π : P → B is a principle Diff(M,U∞)-bundle and f : P → Fr(M) is a Diff(M,U∞)-equivariant map. Associated to π : P → B, we have a (M,U∞)-bundle E → B. The map f : P → Fr(M) corresponds to a vertical framing on E by taking the composition

1×Gf ev d T (M \ ∞) ×G P −−−−→ T (M \ ∞) ×G Fr(M) −→ R × (M \ ∞) where ev((m, v) ×G τ) = τ(m, v) is the evaluation map. More generally, given smooth compact manifolds X ⊂ W and a trivialization τ of TW near X, the space

B^Diff(W, X; τ) := EDiff(W, X) ×Diff(W,X) Fr(W, X; τ) is a model for the classifying space of framed (W, X)-bundles. Here Fr(W, X; τ) is the space of framings on W that agree with τ|X on X. 3 2. Graph Complex Later (Theorem 5.1), we will see that the cycle condition for an element in the graph complex corresponds to the cocycle condition in the deRham complex for the associated characteristic class. All graphs are assumed to be finite, connected, and of valence at least 3. Let Γ be such a graph and denote by V (Γ) the set of vertices of Γ and E(Γ) the set of edges. For each edge e ∈ E(Γ) let + − H(e) = {e , e } be the set of half-edges of e. Choose a labeling of the vertices v1, . . . , vn of Γ and of the edges e1, . . . , em. Definition 6. An orientation of Γ is an orientation of the vector space M RV (Γ) ⊕ RH(e) e∈E(Γ) An orientation of Γ is therefore a choice of ordering on the vertices V (Γ) together with a choice of order of e+ and e− for each e ∈ E(Γ).

Deﬁnition 7. The graph complex is the chain complex (Gn,m, d) where •Gn,m is the Q vector space generated by oriented labeled graphs (Γ, o) with n vertices and m edges, under the equivalence relation (Γ, −o) ∼ −(Γ, o) where −(Γ, o) is obtained from (Γ, −o) by reversing one of the edge orientations. • the diﬀerential d: Gn,m → Gn−1,m−1 is given by the formula X d(Γ, o) := (Γ/e, oe) e∈E0(Γ) where Γ/e is the graph obtained from Γ by collapsing the edge e and oe is the induced orientationm (see §2.2 of [4] for details) and the sum is taken over those edges e ∈ E0(Γ) between distinct vertices; i.e., not loops.

Example. Any graph Γ with a simple loop is identiﬁed with 0 in Gn,m. Example. The graph

has d(Γ) = 0 since Γ/e has a simple loop for every edge e of Γ.

3. Compactifications of Configurations Spaces

Recall that the ordered conﬁguration space Cn(M) of n points in M is a the complement of the fat diagonal in M n, n Cn(M) = {(x1, . . . , xn) ∈ M : xi 6= xj, i 6= j} From the inclusion M \ ∞ ,→ M we have maps n Cn(M, ∞) := Cn(M \ ∞) → Cn(M) ⊂ M n We can identify Cn(M, ∞) as the complement of a subspace Σ in M where n Σ = {(x1, . . . , xn) ∈ M : xi = xj for some i 6= j or xi = ∞ for some i}

There is a natural filtration on Σ by how far a point (x1, . . . , xn) ∈ Σ is from being a configuration. Explicitly, we have a filtration

Σ = Σn ⊃ Σn−1 ⊃ · · · ⊃ Σ1 ⊃ Σ0 4 with ( n )

n [ Σj = (x1, . . . , xn) ∈ M : {xi} ∪ {∞} ≤ j + 1 i=1

Example. The subspace Σ0 is a single point {(∞,..., ∞)}.

Example. The subspace Σ1 is the union over A ⊂ {1, . . . , n} of the subspaces n {(x1, . . . , xn) ∈ M : xi = xj if i, j ∈ A and xk = ∞ if k∈ / A}

The Fulton-MacPherson-Kontsevich compactiﬁcation Cn(M, ∞) of Cn(M, ∞) is obtained by taking successive blow-ups of M n along this ﬁltration. Recall that a blow-up replaces a submanifold with the total space of its normal sphere bundle, viewed as the boundary of a tubular neighborhood. We denote the blow-up of a manifold A along a submanifold B by Bl(A, B).

2 Example. The compactiﬁcation C2(M, ∞) is obtained by blowing up M at (∞, ∞) and then blowing up the result along the image ΣM of the subset

ΣM = M × {∞} ∪ {∞} × M ∪ ∆M in Bl(M 2, {(∞, ∞)}) where ∆ is the diagonal. Since Bl(X,Y ) ' X \ Y , we have 2 C2(M, ∞) = Bl(Bl(M , {(∞, ∞)}), ΣM ) 2 ' Bl(M \{(∞, ∞)} , ΣM ) 2 ' M \{(∞, ∞)}\ ΣM

= C2(M, ∞) The following will be used in §4.

d−1 Lemma 3.1. The space C2(M, ∞) has the homology of S . Hence C2(M, ∞) has the homology of Sd−1.

Note that C2(M, ∞) is not a homology (d − 1)-sphere since it is not compact and C2(M, ∞) is not a homology (d − 1)-sphere since it has nonempty boundary.

Proof Sketch. The projection map C2(M, ∞) → M \ ∞ sending (x1, x2) → x2 is a fibration with fiber M \ {∞, p} over p ∈ M \ ∞. One may show that the fundamental group π1(M \ ∞) acts 2 trivially on H∗(M \ {∞, p}) and that the Serre spectral sequence collapses at the E -term. Thus ∼ ∼ d−1 H∗(C2(M, ∞)) = H∗(M \ {∞, p}) ⊗ H∗(M \ ∞) = H∗(S ) 3.0.1. Stratification of the Boundary. In the proof of Theorem 5.1 below, we will need an explicit description of a stratification of ∂Cn(M, ∞). The stratification is given by assigning to each point x in ∂Cn(M, ∞) a bracketing of {1, . . . , n} with i brackets where numbers l, k are bracketed together if x (viewed as a point in the normal sphere bundle over Σi) lives over a point (x1, . . . , xn) ∈ Σi with xl = xk.

Example. The stratiﬁcation on ∂C2(M, ∞) is into faces S∞×∞, S∞×M , SM×∞, and S∆M corresponding to the brackets (12∞), (1∞)2, (2∞)1, and (12)∞, respectively. The codimension one strata of this stratiﬁcation come from points in the boundary living in the normal sphere bundle of Σ1. These correspond to bracketings with a single bracket. Codimension + one strata of ∂Cn(M, ∞) are in bijection with nonempty subsets A of {1, . . . , n, ∞} =: [n] . For 5 example, if A ⊂ [n]+ does not contain ∞, then A corresponds to the blow-up along the closure of the subset ×n ∆A := (x1, . . . , xn) ∈ (M \ ∞) : xk = xj if k, j ∈ A, otherwise distinct . If Z ⊂ [n]+ does contain ∞, then Z corresponds to the the blow-up along the closure of the subset ∞ ×n ∆Z := (x1, . . . , xn) ∈ M : xj = ∞ if k ∈ Z, otherwise distinct .

+ local d For A ⊂ [n] , let SA ⊂ ∂Cn(M, ∞) denote the face corresponding to A. Let C|A| (R ) denote the ﬁber of SA over a point of ∆A.

local d Lemma 3.2. The space C|A| (R ) is the Fulton-MacPherson-Kontsevich compactification of the configuration space local d d C|A| (R ) := C|A|(R )/(translation, dilation) local d d Moreover, C|A| (R ) is homotopy equivalent C|A|(R ). In particular, if |A| = 2, the Gauss map local d k defines a homotopy equivalence p: C|A| (R ) → S where k = (|A| − 1)d − 1.

4. Propagators

Given a (DM , ∂)-bundle π : E → B, let Cn(π): ECn(π) → B denote the associated Diff(DM , ∂)- bundle with fiber Cn(M \ ∞). Explicitly, if P → B is the principal Diff(M,U∞)-bundle associated to π, then

ECn(π) := P ×Diﬀ(M,U∞) Cn(π)

Similarly, for Y (M, ∞) a Diﬀ(M,U∞)-space, let EY (π) denote the Y (M, ∞)-bundle associated to π. We would like to construct a map ﬁb d−1 ∂ EC2(π) → S

Given a vertical framing τE of π, we can construct such a map (which we will denote p(τE)) by d−1 constructing a map p(κ): ∂C2(M, ∞) → S on the fibers and gluing using the trivialization. We define the map p(κ) componentwise using the stratification of ∂C2(M, ∞) into the blow-up faces

S∞×∞, S∞×M , SM×∞, and S∆M . Details of this construction can be found in §2.2 of [3] where τM is used for κ. x2−x1 On S∞×∞, deﬁne p(κ) by the Gauss map (x1, x2) 7→ . Explicitly, a point x in S∞×∞ lies kx2−x1k on the sphere bundle of the normal bundle of (∞, ∞) ∈ M 2. Since the blow-up only changes things near (∞, ∞), we can view x as a point in ∂Bl(U∞ × U∞, (∞, ∞)). Under the diﬀeomorphism

h∞ × h∞ : U∞ × U∞ → V∞ × V∞ we can view x as a point in ∂Bl((Rd t {∞})×2, (∞, ∞)). Now we have an identification d ×2 ∼ d d ∂Bl((R t {∞}) , (∞, ∞)) = ((R × R ) \{(0, 0)})/R>0 where R>0 acts diagonally. Thus, we can view x ∈ S∞×∞ as the class of a point (x1, x2) 6= (0, 0) in d d d−1 R × R . Under this identification, we define p(κ): S∞×∞ → S by x − x p(κ)(x) = 2 1 k x2 − x1 k

On S∞×M , we deﬁne p(κ) to be the composite

∼ pr1 h∞ d ∼ d−1 S∞×M = ∂Bl(M, ∞) × Bl(M, ∞) −−→ ∂Bl(M, ∞) −−→ ∂Bl(S , ∞) = S

The map p(κ) is defined similarly on SM×∞. 6 Lastly, we define p(κ) on S∆m by the composite ∼ d pr1 d ∼ d−1 S∆M = ∂Bl(R , {0}) × Bl(M, ∞) −−→ ∂Bl(R , {0}) = S d−1 d−1 d−1 Definition 8. The volume form volSd−1 ∈ ΩdR (S ) on S is the SO(d)-invariant closed (d − 1)-form on Sd−1 ⊂ Rd defined by d 1 X i−1 vol d−1 := (−1) xidx1 ∧ · · · ∧ dxci ∧ · · · ∧ dxd S vol(Sd−1) i=1 d−1 Definition 9. A propagator is a closed form ω ∈ ΩdR (EC2(π)) such that ∗ ω| fib = p(τ ) vol d−1 ∂ EC2(π) E S The following lemma says that propagators exist in the context we care about. d−1 Lemma 4.1. A propagator ω ∈ ΩdR (EC2(π)) exists for every vertically framed (Dm, ∂)-bundle π : E → B. Such a propagator is unique up to homotopy. Proof. It suffices to show that the inclusion induced map d−1 d−1 fib H (EC2(π)) → H (∂ EC2(π))(2) is an isomorphism. We have a relative fibration fib (C2(M, ∞), ∂C2(M, ∞)) → (EC2(π), ∂ (EC2(π)) → B By Poincaré-Lefshetz duality and Lemma 3.1, we have q ∼ ∼ d−1 H (C2(M, ∞), ∂C2(M, ∞)) = H2d−q(C2(M, ∞)) = H2d−p(S ) p,q which vanishes for 0 ≤ q ≤ d. Thus the E2-terms E2 in the Serre spectral sequence for the above relative fibration vanish for 0 ≤ p + q ≤ d. Hence n fib H (EC2(π), ∂ EC2(π)) = 0 fib for n = d − 1, d. Using the long exact sequence for the pair (EC2(π), ∂ (EC2(π)), we see that the map (2) is an isomorphism. For the uniqueness statement, see [3, Lemma 3]. 5. Kontsevich’s Characteristic Classes We give a definition of Kontsevich’s characteristic classes for framed sphere bundles as in §1. More explicitly, let M be an homology d-sphere with d odd and π : E → B be a (DM , ∂)-bundle. ∗ We will define classes I(Γ) ∈ ΩdR(B) for each oriented graph Γ. The definition we will give uses a fixed propagator ω; however, we will see (Theorem 5.1) that the resulting class I(Γ) does not depend on the choice of ω. Consider the associated bundle Cn(π) with fiber Cn(M \ ∞). The propagator ω is a deRham form on EC2(π), not on ECn(π). We therefore need way to get from ECn(π) to EC2(π). Such a map will be defined as a projection in terms of H(e) Γ. For each edge e of an oriented graph (Γ, o) ∈ Gn,m, choose an orientation of R so that the vector space M RV (Γ) ⊕ RH(e) e∈E(Γ) with this orientation on RH(e) and the total ordering ρ of the vertices, is o.

Deﬁnition 10. For an edge e of Γ, deﬁne φe : ECn(π) → EC2(π) to be the projection

(b, (x1, . . . , xn)) 7→ (b, (xk, xj)) where e is the edge between vertices vk and vj using the labels from ρ. 7 For each oriented graph Γ with n vertices and m edges, we obtain an m(d − 1) − nd form on B by looking at the image of ω under the following composite

V φ∗ e R ∗ e∈E(Γ) ∗ fiber ∗ ∗ ΩdR(EC2(π)) −−−−−−→ ΩdR(ECn(π)) −−−→ ΩdR(B) −→ H B V ∗ R Deﬁnition 11. Let ω(Γ) = φe(ω) and I(Γ) = ﬁber ω(Γ). e∈E(Γ)

Theorem 5.1 (Kontsevich). Let Γ be a cycle in Gn,m. (i) I is a chain map: dI(Γ) = (−1)|I(dΓ)|I(dΓ). In particular, dI(Γ) = 0. (ii) The class [I(Γ)] ∈ H∗(B; R) is independent of the choice of propagator ω. (iii) The class [I(Γ)] is natural with respect to bundle maps of framed bundles. (iv) Evaluation of I(Γ) on base spaces gives a homomorphism

Ωm(d−1)−nd(B^Diﬀ(DM , ∂)) → R

where Ω∗(−) is the oriented bordism group functor.

Remark. The class I(Γ) may depend on the choice of vertical framing τE of the (DM , ∂)-bundle π : E → B. See, for example, [3, Rmk 2]. Proof of (iii) and (iv). This follows from the facts that integration along the ﬁbers is natural with respect to bundle maps and that propagators pullback to propagators along bundle maps of framed bundles. Part (iv) is obvious. The proofs of parts (i) and (ii) of Theorem 5.1 will use the following. Recall from §3.0.1, that + codimension one strata of ∂Cn(M, ∞) are in bijection with subsets A ⊆ [n] := {1, . . . , n, ∞} of size at least 2. Let [n] denote the subset {1, . . . , n} of [n]+.

Deﬁnition 12. For Γ ∈ Gn,m and A ⊆ [n] with |A| ≥ 2, let ΓA be the subgraph of Γ consisting of vertices va for a ∈ A and all of the edges connecting them. Let Γ/A be the quotient graph Γ/(ΓA) where all vertices va, a ∈ A are collapsed to a single vertex [∗] and all edges e ∈ ΓA are collapsed to [∗].

Example. Consider the Θ graph Γ with two vertices v1 and v2 and three edges connecting them. Let A = {1, 2}. Then ΓA = Γ and Γ/A is a point. For A ⊂ [n]+ let Ac denote the complement of A in [n]+. Note that if ∞ ∈ A then Ac ⊂ [n]. Let πA : ESA(π) → B be the SA-bundle corresponding to π. The framing τE on π determines a natural diﬀeomorphism local d fA : ESA(π) → C|A| (R ) × ECn−|A|+1(π) d−1 Let Γ ∈ Gn,m and ω ∈ ΩdR (EC2(π)) a propagator. If A ⊂ [n], we have ∗ ∗ ∗ (3) ω(Γ)|ESA(π) = fA pr1ωˆ(ΓA) ∧ pr2ω(Γ/A) If A ⊂ [n]+ and ∞ ∈ A, then we have ∗ ∗ c ∗ c (4) ω(Γ)|ESA(π) = fA pr1ωˆ(Γ/A ) ∧ pr2ω(ΓA )

∗ ∗ local d Here ω(Γ/A) ∈ ΩdR(ECn−|A|+1(π)) is defined by Definition 11 andω ˆ(ΓA) ∈ ΩdR(C|A| (R )) is local d d−1 ∗ defined as follows: Let p: C2 (R ) → S be the Gauss map. Writeω ˆ = p (volSd−1 ) for the pullback of the volume form. 8 ∗ local d Definition 13. Defineω ˆ(ΓA) ∈ ΩdR(C|A| (R )) to be the wedge of forms ^ ˆ∗ ωˆ(ΓA) = φeωˆ

e∈E(ΓA)

ˆ local d local d where φe : C|A| (R ) → C2 (R ) picks out the two coordinates corresponding to the vertices of e. Remark. We explain the difference in the formulas (3) and (4). If A ⊂ [n]+ does not contain ∞, then the n − |A| + 1 points in M \ ∞ in a fiber of ECn−|A|+1(π) come from picking out the n − |A| + points xk for k ∈ [n] \ A \ {∞} and an additional point xj chosen to represent the |A| points xj, j ∈ A that are all equal. The corresponding graph therefore has a vertex vk for each k ∈ [n] \ A and a single vertex [∗] representing all of the vertex vj, j ∈ A. This is exactly Γ/A. On the other hand, + if A ⊂ [n] contains ∞, then the n − |A| + 1 points in M \ ∞ in a fiber of ECn−|A|+1(π) come from + picking out the the n − |A| + 1 points xk for k ∈ [n] \ A. The corresponding graph is therefore ΓAc , which does not contain a point representing all of vj, j ∈ A. Heuristically, if ∞ ∈ A, then not only did the xj, j ∈ A all collide, they also disappeared at ∞. R For the proof of (i), we need to compute dI(Γ). Since I(Γ) = Fiber ω(Γ) is an integral of a form, we would like to use Stoke’s theorem to compute dI(Γ). Stoke’s theorem for integration along ∗ the fibers takes the following form. Let ρ: X → Y be a bundle. For γ ∈ ΩdR(Y ) a form, let J(γ) = (−1)deg(γ)γ. Integration along the fiber satisfies the following generalized Stoke’s theorem: Z Z Z d α = dα + J α Fib(ρ) Fib(ρ) ∂Fib(ρ) In particular, if α is a closed form, we have Z Z (5) d α = J α Fib(ρ) ∂Fib(ρ)

V ∗ Proof of (i). Since ω(Γ) = φe(ω) is the wedge of closed form, it itself is closed. We apply (5) e∈E(Γ) to the form α = ω(Γ) to get Z Z Z dI(Γ) = d ω(Γ) = dω(Γ) + J ω(Γ) Fib(Cn(π)) Fib(Cn(π)) ∂Fib(Cn(π)) The second term in the sum can be rewritten as Z X Z X Z J ω(Γ) = J ω(Γ) + ω(Γ) ∂Fib(Cn(π)) Fib(πA) Fib(πZ ) A⊂[n] Z⊂[n]+ |A|≥2 |Z|≥2,∞∈Z using the description of the boundary of the ﬁber in terms of its strata, a ∂Fib(Cn(π)) = ∂Cn(M, ∞) = Fib(πA) t ( lower dimensional strata)

A⊂[n]+ |A|≥2

To show that dI(Γ) = (−1)|I(dΓ)|I(dΓ) we are therefore left with showing that X Z ω(Γ) = I(dΓ)(6) Fib(Cn(πA)) A⊂[n]+ |A|≥2

9 We have deﬁned dΓ to be the formal sum over edges of Γ of the oriented graphs (Γ/e, oe) obtained by collapsing an edge e of Γ with the induced orientation. Using the labeling of vertices of Γ by {1, . . . , n}, we can rewrite dΓ as X dΓ = (Γ/A, oA) A∈E(Γ) where the sum is taken over the collection E(Γ) of all subsets A ⊆ [n] with |A| = 2 and so that the corresponding pair of vertices in Γ are connected by a single edge. For A ∈ E(Γ), we have used A to denote both a pair of integers 1, . . . , n and the corresponding edge of Γ. By Lemmas 5.2, 5.6, 5.4 and Corollaries 5.3, and 5.5 below, the integral along a ﬁber of πA of ω(Γ) vanishes unless |A| = 2 and (A) if ∞ ∈ A, then Γ/Ac is a graph with two vertices and at most 1 edge, or (B) if ∞ ∈/ A, then ΓA is a graph with two vertices and a single edge. However, situation (A) cannot occur. Indeed, say |A| = 2 with ∞ ∈ A. Then A = {k, ∞} for some c k ∈ {1, . . . , n}. The graph Γ/A consists of the vertex vk and a vertex [∗] coming from the collapse c c of A . Edges of Γ/A are in bijection with edges of Γ involving vk. By assumption, every vertex of Γ has valence at least 3. Thus Γ/Ac has at least 3 edges. Thus the only contribution to the integral of ω(Γ) along ∂Fib(Cn(π)) comes from situation (B). We therefore just need to show that for A ∈ E(Γ), the following equality holds Z ω(Γ) = I(Γ/A, oA) Fib(πA) for A ∈ E(Γ). Assuming for a moment that the orientations work out, we have from (3) and Lemma 3.2, Z Z Z ω(Γ) = ωˆ(ΓA) ∧ ω(Γ/A) local d Fib(πA) C2 (R ) Fib(Cn−1(π)) Z Z = volSd−1 ∧ ω(Γ/A) d−1 S Fib(Cn−1(π)) Z = ω(Γ/A) Fib(Cn−1(π)) = I(Γ/A)

Details on the orientations can be found in the proof of Lemma A.2 of [4].

There are three ways a nonempty subset A ⊆ [n]+ can fail to be in E(Γ): • every vertex of ΓA has valence ≥ 2, • |A| ≥ 3 and every vertex of ΓA has valence 1, or • |A| = 2 and the corresponding vertices of Γ are not connected.

Lemma 5.2. Let A ⊆ [n] with |A| ≥ 2. If every vertex of ΓA has valence ≥ 2, then the integral along the ﬁber of πA of ω(Γ) vanishes, Z ω(Γ) = 0 Fib(πA) Proof. We break the proof into two cases: A) All vertices of ΓA have valence at least 3. 10 B) ΓA has a vertex of valence 2. Case A. By (3), it suﬃces to show that the integral Z ωˆ(ΓA) local d C|A| (R )

local d vanishes. To do this, we will show that ω(ΓA) is not a top form on C|A| (R ): local d dim C|A| (R ) 6= deg ω(ΓA)

local d 0 0 The dimension of C|A| (R ) is (|A| − 1)d − 1 and the degree of ω(ΓA) is m (d − 1) where m is the number of edges of ΓA. Since each of the |A| vertices of ΓA has valence at least 3, we have 2m0 ≥ 3|A|. In particular, m0 ≥ |A| > |A| − 1. Thus (|A| − 1)d − 1 > m0(d − 1) and therefore the integral vanishes. Case B. Let a be a vertex of ΓA of valence 2. Say ΓA has two edges eb and ec with boundary vertices {a, b} and {a, c}, respectively. Deﬁne an automorphism ι of ESA(π) sending ( xb + xc − xa k = a ι(xk) = xk k 6= a

∗ local d Then ι ω(Γ) = ω(Γ) and ι reverses the orintation of Cj (R ). Thus the integral vanishes. Using (4) instead of (3), the exact same argument applied to Γ/Zc for Z ⊂ [n]+ containing ∞ gives the following: Corollary 5.3. Let Z ⊆ [n]+ with |Z| ≥ 2 and ∞ ∈ Z. If every vertex of Γ/Zc has valence ≥ 2, then the integral along the ﬁber of πZ of ω(Γ) vanishes, Z ω(Γ) = 0 Fib(πZ )

Lemma 5.4. Let A ⊆ [n] with |A| ≥ 3. If ΓA has a vertex of valence 1, then the integral Z ω(Γ) = 0 Fib(πA) vanishes.

Proof. Say b is a vertex of ΓA of valence 1. Let e be the unique edge of ΓA that is connected to local d b. Say ∂e = {a, b}. Write j = |A|. Let Y be the subset of Cj (R ) of points whose xa and xb coordinates are equal. Consider the map

local d q : Cj (R ) → Y deﬁned by ( xk k 6= a, b q(xk) = xb−xa k = a, b kxb−xak (d−1)m0 Then there exists some form α in ΩdR (Y ) so that ∗ q α = ω(ΓA)| local d Cj (R )

local d Since Y has dimension strictly less than the dimension of Cj (R ), the integral vanishes. 11 Again we can apply the same argument to Γ/Zc for Z ⊂ [n]+, |Z| ≥ 3 containing ∞. Corollary 5.5. Let Z ⊂ [n]+ containing ∞. If |Z| ≥ 3 has a vertex of valence 1, then the integral Z ω(Γ) = 0 Fib(πZ ) vanishes. Lemma 5.6. Let A ⊆ [n] with |A| = 2. If the corresponding two vertices of Γ are not connected by any edge then the integral along the ﬁber of ω(Γ) vanishes, Z ω(Γ) = 0 Fib(πA)

Proof. By (3) it suﬃces to show thatω ˆ(ΓA) = 0. Since the vertices of Γ corresponding to A are not connected, the graph ΓA has no edges. By deﬁnition,ω ˆ(ΓA) is a wedge of forms, ^ ˆ∗ ωˆ(ΓA) = φeωˆ

e∈E(ΓA)

Since ΓA has no edges, this is a wedge over an empty set, and hence zero. We now proof part (ii) of Theorem 5.1.

Proof of (ii). Let ω0 and ω1 be two propagators corresponding to the same homotopy class of the framing τE. We want to construct a form α so that the following equality holds Z Z ω0(Γ) − ω1(Γ) = dα ﬁber ﬁber

By Lemma 4.1, there exists a formω ˜ on [0, 1] × EC2(π) restricting to ωi, i = 0, 1 on {i} × EC2(π). V ∗ Writeω ˜(Γ) for (Id ⊗ φe)(˜ω). By the generalized Stoke’s theorem (5) for closed forms such as e∈E(Γ) ω˜(Γ), we have Z Z d ω˜(Γ) = J ω˜(Γ) [0,1]×Fib(Cn(π)) ∂([0,1]×Fib(Cn(π)))

Now the boundary ∂([0, 1] × Fib(Cn(π))) can be rewritten as the disjoint union {0} × Fib(Cn(π)) t {1} × Fib(Cn(π)) t [0, 1] × ∂Fib(Cn(π)) where the second term has the opposite orientation. Thus Z Z Z Z d ω˜(Γ) = J ω0(Γ) − ω1(Γ) + ω˜(Γ) [0,1]×Fib(Cn(π)) Fib(Cn(π)) Fib(Cn(π)) [0,1]×∂Fib(Cn(π)) It suffices to show that the third term in the sum vanishes. This follows from a similar argument as above. References [1] William Fulton and Robert MacPherson. A compactification of configuration spaces. Ann. of Math. (2), 139(1):183–225, 1994. [2] Maxim Kontsevich. Feynman diagrams and low-dimensional topology. In First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 97–121. Birkhäuser,Basel, 1994. [3] Tadayuki Watanabe. On Kontsevich’s characteristic classes for higher dimensional sphere bundles. I. The simplest class. Math. Z., 262(3):683–712, 2009. [4] Tadayuki Watanabe. On Kontsevich’s characteristic classes for higher-dimensional sphere bundles. II. Higher classes. J. Topol., 2(3):624–660, 2009.