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A four-dimensional boundary case is outside the smooth h-cobordism theorem

Statement refuted

False claim: every compact smooth h-cobordism over a closed simply connected manifold of dimension four is diffeomorphic to the product. More precisely: the dimension hypothesis $n \geq 5$ of the smooth h-cobordism theorem (The smooth simply connected h-cobordism theorem) can be weakened to $n = 4$ , so that every h-cobordism of boundary dimension four over a closed simply connected 4-manifold is trivial.

Facts & Assumptions

Given: The dimension count for a clean embedded Whitney disk in a middle level of dimension $m$ : the sheets must satisfy $a, b \leq m - 3$ , or one of them has dimension at least $3$ in the borderline version; and the literature record for boundary dimension four.

[F1]

In boundary dimension four the relevant middle level is four-dimensional and the general-position count fails: for complementary sheets with $a = b = 2$ one has $2 + a - m = 0$ , and the smooth Whitney trick has no general embedded-disk replacement; the failure is the content of the dimension-four Whitney-trick remark together with its companion counterexample (The smooth Whitney trick fails in dimension four).

[F2]

The conclusion fails in the smooth category: there are smooth orientable simply connected 4-manifolds that are all smoothly s-cobordant and homeomorphic but pairwise not diffeomorphic (Kasprowski--Powell--Ray, EMS Surv. Math. Sci. 9 (2022), Example 1.13, first pair due to Donaldson), so no h-cobordism between two of them is a product; equivalently the smooth h-cobordism theorem is false for boundary dimension four in general, while the topological statement holds for good fundamental groups, in particular for the trivial group, by Freedman (The h-cobordism theorem does not cover boundary dimension four).

[F3]

An h-cobordism is a compact smooth cobordism whose two face inclusions are homotopy equivalences, and its triviality is the existence of a diffeomorphism to the product relative to the incoming boundary (h-Cobordism, Simply connected topological spaces).

[F4]

The smooth simply connected h-cobordism theorem assumes boundary dimension $n \geq 5$ , equivalently total dimension $n + 1 \geq 6$ (The smooth simply connected h-cobordism theorem).

Counterexample

1.1F1given

The proof route fails in boundary dimension four: by [F1], a clean embedded Whitney disk in a four-dimensional middle level cannot generally be produced, since the general-position dimension count for complementary sheets of dimensions $2$ and $2$ gives no room for an embedded disk; the Whitney-trick step of the theorem is therefore unavailable exactly when $n = 4$ .

2.1F2F3givenstep 1.1

The conclusion also fails, so the failure is not merely a gap in the proof: by [F2] there are smooth orientable simply connected 4-manifolds that are smoothly s-cobordant and homeomorphic but not diffeomorphic, and an s-cobordism is in particular an h-cobordism between them; were such an h-cobordism diffeomorphic to the product relative to its incoming boundary, [F3] would make its two faces diffeomorphic, contradicting the choice of the pair.

3.1F2F4step 2.1∎

Therefore the hypothesis $n \geq 5$ of the smooth h-cobordism theorem cannot be weakened to $n = 4$ : the dimension count at the Whitney step stops working and the conclusion itself is false in general for simply connected closed 4-manifolds, while the parallel topological statement for good fundamental groups is a genuinely different theorem. No smooth four-dimensional Poincaré or disk conclusion follows from the theorem of this page, whose statement is restricted to boundary dimension at least five by [F4].

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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)
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)
Daniel Kasprowski, Mark Powell and Arunima Ray, Counterexamples in 4-manifold topology, EMS Surveys in Mathematical Sciences 9 (2022) 193--249 (standard reference, not scraped)