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Homotopy spheres of dimension at least five bounding a contractible manifold are standard spheres

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a compact contractible smooth $(n + 1)$ -manifold with connected boundary $Σ = \partial X$ , where $n \geq 5$ and $Σ$ is a homotopy $n$ -sphere (equivalently by Hurewicz and Whitehead: $Σ$ is simply connected and $H_{*} (Σ; Z) ≅ H_{*} (S^{n}; Z)$ ) (Simply connected topological spaces, Absolute Hurewicz theorem at the first nonzero degree, Whitehead theorem). Then $X$ is diffeomorphic to the disk $D^{n + 1}$ , and $Σ$ is diffeomorphic to $S^{n}$ . Consequently a smooth homotopy $n$ -sphere, $n \geq 5$ , that bounds a compact contractible smooth manifold is diffeomorphic to the standard sphere, and no exotic sphere in these dimensions bounds a contractible manifold.

Facts & Assumptions

Given: A compact contractible smooth $(n + 1)$ -manifold $X$ with connected boundary $Σ = \partial X$ a homotopy $n$ -sphere, $n \geq 5$ ; the Axiom of Choice.

[F1]

A nonempty contractible space has the singular homology of a point in every degree and coefficient group (Contractible nonempty spaces have the homology of a point), and the relative chain complexes are free, so the cohomological universal coefficient sequence computes $H^{j}$ from the homology (The cohomology universal-coefficient sequence splits nonnaturally).

[F2]

Under AC, for every numerable real bundle $E \to B$ of rank $n \geq 0$ over a CW complex or an admissible base (a paracompact Hausdorff CGWH space of CW homotopy type), $w_{1} (E) = 0$ if and only if $E$ is orientable; applied to the tangent bundle, this says $X$ is orientable since $w_{1} (T X) \in H^{1} (X; Z / 2) = 0$ (The first Stiefel–Whitney class classifies orientability).

[F3]

If $X = U \cup V$ is a two-set van Kampen cover with path-connected members and simply connected overlap, then $π_{1} (U) * π_{1} (V) ≅ π_{1} (X)$ ; for $n \geq 2$ the sphere $S^{n}$ is simply connected (A simply connected overlap turns the van Kampen pushout into a free product, $S^{n}$ is simply connected for every $n \geq 2$ , Simply connected topological spaces).

[F4]

Excision: if $Z \subseteq X$ has closure contained in the interior of $A$ , then $H_{*} (X - Z, A - Z; G) ≅ H_{*} (X, A; G)$ for all degrees and coefficients, and the long exact sequence of a pair computes relative groups from acyclic absolute groups (Excision for singular homology, Long exact sequence of a pair).

[F5]

For a compact $Z$ -oriented manifold whose boundary is a disjoint union of two closed boundary manifolds $A ⊔ B$ , cap with the relative fundamental class gives $H^{p} (W, A; Z) ≅ H_{n + 1 - p} (W, B; Z)$ (Fully relative Poincaré–Lefschetz duality).

[F6]

Compact smooth manifolds have finite CW homotopy models (A handle decomposition gives a relative CW complex). On these models, the relative homotopy exact sequence and relative Hurewicz convert a homology equivalence between simply connected spaces into a weak equivalence, and Whitehead makes it a homotopy equivalence under the assumed AC (Long exact sequence of relative homotopy groups); an h-cobordism is a cobordism whose two face inclusions are homotopy equivalences (Homotopy equivalences induce isomorphisms on singular homology, Relative Hurewicz theorem in the simple-connectivity range, Whitehead theorem, Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).

[F7]

Every smooth manifold with boundary has a smooth collar, and a disk glued to a product along its boundary sphere restores the disk: $D^{n + 1} ≅ (S^{n} \times [0, 1]) \cup_{S^{n} \times {0}} D^{n + 1}$ (Collar neighborhood theorem, Smooth collars of a manifold boundary).

[F8]

The h-cobordism theorem identifies every h-cobordism over a closed simply connected $n$ -manifold with the product for $n \geq 5$ (The smooth simply connected h-cobordism theorem).

Proof

technique · direct

1.1F1given

Choose an embedded closed $(n + 1)$ -disk $D^{n + 1} \subset int X$ (Smooth embeddings) and put $W : = X ∖ int D^{n + 1}$ ; then $W$ is a compact smooth $(n + 1)$ -manifold with boundary the disjoint union $Σ ⊔ S^{n}$ of the given boundary and the boundary sphere of the removed disk, and $W$ is connected: remove a point at the disk center, reroute paths locally around that point in dimension $n + 1 \geq 6$ , then radially retract the punctured disk onto its boundary. The same construction retains compactness and the smooth boundary collars.

1.2F1F2F6given

The compact smooth $X$ has CW homotopy type by [F6], and a finite chart cover with a subordinate smooth partition (Smooth partitions of unity exist on manifolds) supplies a numeration of its tangent bundle. Thus [F2] applies to this bundle. $X$ is orientable: $H_{*} (X; Z / 2) = H_{*} (p t; Z / 2)$ by [F1], so the universal coefficient sequence gives $H^{1} (X; Z / 2) = 0$ (both $Hom (H_{1} (X), Z / 2)$ and $Ext (H_{0} (X), Z / 2)$ vanish because $H_{1} (X) = 0$ and $H_{0} (X) = Z$ is free); hence $w_{1} (T X) = 0$ and by [F2] the tangent bundle $T X$ is orientable, so $X$ carries an orientation and $W$ inherits the restricted orientation.

2.1F3given

$π_{1} (W) = 1$ : use the open cover $U = X ∖ D_{0}$ , $V = int D^{n + 1}$ , where $D_{0}$ is a smaller concentric closed disk. Radial compression retracts $U$ onto $W$ , $V$ is contractible and $U \cap V ≅ S^{n} \times (a, 1)$ is simply connected by [F3]. Both open members are path connected by step 1.1 and radial compression. Van Kampen therefore gives $π_{1} (X) ≅ π_{1} (W) * π_{1} (D^{n + 1}) = π_{1} (W)$ , and $π_{1} (X) = 1$ because $X$ is contractible, so $π_{1} (W) = 1$ .

2.2F4step 1.1

$H_{*} (W, S^{n}; Z) = 0$ : let $D_{0} \subset int D^{n + 1}$ be a smaller closed concentric disk; then excision [F4] with $Z = D_{0}$ and $A = D^{n + 1}$ gives $H_{*} (X - D_{0}, D^{n + 1} - D_{0}; Z) ≅ H_{*} (X, D^{n + 1}; Z) = 0$ , the last vanishing because $X$ and $D^{n + 1}$ are acyclic and [F4] computes the pair by its long exact sequence; the pair $(X - D_{0}, D^{n + 1} - D_{0})$ deformation retracts through a collar onto $(W, S^{n})$ , so $H_{*} (W, S^{n}; Z) = 0$ as well.

3.1F1F5step 1.2step 2.2

$H_{*} (W, Σ; Z) = 0$ : by step 1.2 the manifold $W$ is compact and $Z$ -oriented with boundary the disjoint union $Σ ⊔ S^{n}$ , so [F5] with $A = S^{n}$ , $B = Σ$ gives $H_{k} (W, Σ; Z) ≅ H^{n + 1 - k} (W, S^{n}; Z)$ ; by the universal coefficient sequence [F1] applied to the relative chain complex and step 2.2, the group $H^{n + 1 - k} (W, S^{n}; Z)$ vanishes, so $H_{k} (W, Σ; Z) = 0$ .

4.1F4F6step 2.1step 2.2step 3.1

Both end inclusions induce homology isomorphisms by the pair sequences and steps 2.2–3.1. The sources and $W$ are nonempty simply connected. Transport each map to the finite CW models of [F6] and replace it by a cellular map under AC (Cellular approximation for maps of CW pairs). Its CW mapping-cylinder pair is $1$ -connected and has zero relative integral homology. Inductively, if its relative homotopy groups below $j \geq 2$ vanish, relative Hurewicz in [F6] identifies $π_{j}$ with the zero $H_{j}$ ; hence all relative groups vanish. The relative homotopy exact sequence makes the map a weak equivalence, and Whitehead yields a homotopy inverse. Transport it back to the original manifolds. Thus the two actual end inclusions are homotopy equivalences and $W$ is an h-cobordism.

5.1F7F8step 4.1∎

By the h-cobordism theorem [F8] applied to the h-cobordism $W$ with $\dim W = n + 1 \geq 6$ , there is a diffeomorphism $W \to S^{n} \times [0, 1]$ that is the identity on $S^{n}$ ; its restriction to the other face is a diffeomorphism $Σ \to S^{n}$ . Gluing the disk back along the collar by [F7] identifies $X$ with $(S^{n} \times [0, 1]) \cup_{S^{n} \times {0}} D^{n + 1} ≅ D^{n + 1}$ , so $X$ is diffeomorphic to the disk and $Σ$ to $S^{n}$ . This is Milnor's Proposition A of §9.

The full AC hypothesis licenses the cited UCT, orientability, duality, Hurewicz and CW comparison results as stated. The h-cobordism theorem itself uses only countable choice; AC supplies it by restricting a choice function to a countable family. For the parenthetical homology-sphere criterion, absolute Hurewicz gives vanishing homotopy below n and a map $S^{n} \to Σ$ representing a generator of $H_{n}$ ; it is a homology equivalence, so the same CW comparison proves it is a homotopy equivalence.

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Sources

John Milnor, Lectures on the h-Cobordism Theorem (notes by L. Siebenmann and J. Sondow, Princeton University Press 1965; scanned edition with searchable text layer) (standard reference, not scraped)
Wolfgang Lück, A Basic Introduction to Surgery Theory (ICTP lecture notes, 27 October 2004; complete author text) (standard reference, not scraped)