Kirby Problem 4.18

Definitions and mathematical statement

A homotopy equivalence of finite CW complexes is simple if it can be expressed, up to homotopy, by finitely many elementary collapses and their inverses. An elementary collapse removes a cell together with a free face. Compact smooth manifolds carry their canonical simple homotopy types through finite triangulations.

(a) Do there exist closed connected smooth four-manifolds $M,N$ that are homotopy equivalent but not simple homotopy equivalent? The latter condition means that no homotopy equivalence between them is simple; exhibiting one nonsimple equivalence would not suffice if a different simple equivalence exists.

A smooth $h$-cobordism from $M$ to $N$ is a compact smooth five-manifold $W$ with boundary $M\sqcup N$ whose two boundary inclusions are homotopy equivalences. It is an $s$-cobordism when those inclusions are simple homotopy equivalences. With orientations specified, the incoming boundary has the reversed orientation.

(b) Do there exist closed smooth four-manifolds joined by a smooth $h$-cobordism but by no smooth $s$-cobordism?

Again, the nonexistence quantifier ranges over all possible cobordisms between the pair. The failure of one chosen $h$-cobordism to be simple does not establish this conclusion. Both questions require smooth manifolds and, in (b), smooth cobordisms; constructions only in the topological category do not answer them. Fundamental groups are unrestricted, since simple homotopy contains information beyond the ordinary homotopy type when their Whitehead torsion can be nonzero.

Short English statement

Can smooth four-manifolds be homotopy equivalent without being simply equivalent, or h-cobordant without any smooth s-cobordism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2894 (KP-4.18), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.18. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.