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The handle complex of an h-cobordism is contractible over the group ring, with an explicit contraction

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(W; M_{0}, M_{1})$ be a nonempty connected h-cobordism whose inclusions $M_{i} ↪ W$ are homotopy equivalences, with a finite handle decomposition relative to $M_{0}$ , and let $C_{*}^{h} (W, M_{0})$ be its based handle complex over $R = Z [π_{1} (M_{0})]$ . Then $C_{*}^{h} (W, M_{0})$ is contractible: it admits a right $R$ -linear chain contraction $s$ with $d s + s d = i d$ . Choose a homotopy inverse $r : W \to M_{0}$ and homotopies $r i ≃ {i d}_{M_{0}}$ and $i r ≃ {i d}_{W}$ , cellularly approximate $r$ , and lift the maps and homotopies equivariantly to universal covers. The lifted inclusion induces a chain homotopy equivalence, so its algebraic mapping cone is contractible by the lifted-cone lemma. The based pair sequence $0 \to C_{*} ({\tilde{M}}_{0}) \overset{C_{*} (\tilde{i})}{\to} C_{*} (\tilde{W}) \to C_{*}^{h} (W, M_{0}) \to 0$ is degreewise split; its quotient complex is chain homotopy equivalent to that mapping cone, hence is contractible. The argument constructs a contraction from the lifted homotopy inverse and homotopies; it does not infer contractibility from acyclicity.

Facts & Assumptions

Given: An h-cobordism $(W; M_{0}, M_{1})$ with a finite handle decomposition relative to $M_{0}$ , and its based handle complex $C_{*}^{h} (W, M_{0})$ over $R = Z [π_{1} (M_{0})]$ , all lifted data taken from the handle decomposition.

[F1]

Both inclusions of an h-cobordism are homotopy equivalences, so the inclusion $ι : M_{0} ↪ W$ induces an isomorphism on fundamental groups, and any homotopy inverse $r : W \to M_{0}$ with homotopies $r ι ≃ {i d}_{M_{0}}$ and $ι r ≃ {i d}_{W}$ can be replaced by a cellular map and cellular homotopies in the CW structure induced by the handle decomposition (h-Cobordism, Cellular approximation for maps of CW pairs, Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).

[F2]

A lifted cellular homotopy equivalence of connected finite CW complexes induces a right-linear chain homotopy equivalence of the based cellular chain complexes of their universal covers, with chain homotopies induced by the lifted geometric homotopies, and its algebraic mapping cone is contractible (A lifted finite CW equivalence has a contractible group-ring mapping cone, A chain homotopy equivalence, The mapping cone of a chain map, Lifting criterion for maps from path-connected locally path-connected spaces, Universal covering spaces).

[F3]

A chain map of bounded based free complexes is a chain homotopy equivalence if and only if its algebraic mapping cone is contractible, and the based-exact-sequence clause of the AT-22 sum theorem identifies the torsion of the quotient of a degreewise split based exact sequence with the relevant cone torsion (A chain map is a homotopy equivalence exactly when its cone is contractible, Composition and based-pair sum formulas for Whitehead torsion).

[F4]

The based handle complex is the based relative cellular chain complex of the relative CW pair induced by the handle decomposition, and the degreewise split based pair sequence of a based subcomplex and its relative quotient exists with the quotient complex the relative based complex of the pair (The based handle chain complex over the fundamental group ring).

Proof

1.1F1given

Choose a homotopy inverse $r : W \to M_{0}$ of the inclusion $ι$ together with homotopies $H : r ι ≃ {i d}_{M_{0}}$ and $K : ι r ≃ {i d}_{W}$ ; by [F1] the inclusion is a homotopy equivalence and $r$ may be assumed cellular with cellular homotopies, so all this data is compatible with the CW structure induced by the handle decomposition.

2.1F2step 1.1

Lift $r$ and the homotopies to the universal covers; the lifted inclusion $C_{*} (\tilde{ι}) : C_{*} ({\tilde{M}}_{0}) \to C_{*} (\tilde{W})$ is a right $R$ -linear chain homotopy equivalence, with the chain homotopies induced by the lifted geometric homotopies, and its algebraic mapping cone $Cone (C_{*} (\tilde{ι}))$ is contractible by the lifted-cone lemma of [F2].

3.1F4step 2.1

Consider the based pair sequence $0 \to C_{*} ({\tilde{M}}_{0}) \overset{C_{*} (\tilde{ι})}{\to} C_{*} (\tilde{W}) \to C_{*}^{h} (W, M_{0}) \to 0$ of [F4]; it is degreewise based exact and split, because the relative cells of the handle decomposition and the cells over $M_{0}$ together form a basis of $C_{*} (\tilde{W})$ in each degree.

4.1F3F4step 2.1step 3.1

The quotient map $γ : Cone (C_{*} (\tilde{ι})) \to C_{*}^{h} (W, M_{0})$ , $γ (b, a) : = [b]$ , is a chain map for the cone differential of The mapping cone of a chain map, since $q \circ C_{*} (\tilde{ι}) = 0$ and $q$ is a chain map; in the degreewise splitting $C_{*} (\tilde{W})_{n} = C_{*} ({\tilde{M}}_{0})_{n} \oplus C_{n}^{h}$ of [F4] its kernel consists of the pairs $(C_{*} (\tilde{ι}) a^{'}, a)$ and is the complex $K (a^{'}, a) = (\partial a^{'} + a, - \partial a)$ on $C_{*} ({\tilde{M}}_{0})_{n} \oplus C_{*} ({\tilde{M}}_{0})_{n - 1}$ , which the explicit map $s (a^{'}, a) = (0, a^{'})$ contracts, so the kernel is contractible. The displayed sequence $0 \to \ker γ \to Cone (C_{*} (\tilde{ι})) \to C_{*}^{h} (W, M_{0}) \to 0$ is degreewise split; choosing a graded splitting $σ$ and correcting it by the kernel contraction as in the proof of the based-exact-sequence clause of the AT-22 sum theorem produces a chain section $σ^{'}$ with $d σ^{'} = σ^{'} d$ , and then $γ H σ^{'}$ contracts $C_{*}^{h} (W, M_{0})$ for any contraction $H$ of the cone.

5.1F2F3step 2.1step 4.1∎

Therefore $C_{*}^{h} (W, M_{0})$ is contractible, with a right $R$ -linear contraction $s$ constructed from the lifted homotopy inverse and homotopies; in particular the contraction is produced by the geometric data and not inferred from the vanishing of homology.

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Sources

Wolfgang Lück, A Basic Introduction to Surgery Theory (ICTP lecture notes, 27 October 2004; complete author text) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, electronic edition) (standard reference, not scraped)