Spineless 5-manifolds and the deformation conjecture

TL;DR

The paper constructs a spineless 5-manifold M, simple-homotopy equivalent to a wedge of 11 copies of $S^2$, using an exotic smooth structure on a connected sum of $S^2 imes S^2$ to obstruct a 0–2 handle decomposition. It links the existence of spines to the deformation conjecture for 2-complexes (generalizing Andrews–Curtis) and explains that, under the conjecture, simple-homotopy equivalent 5-manifolds with spines would be PL-isomorphic and have diffeomorphic boundaries. A central question is posed: could exotic pairs of 4-manifolds bound 5-manifolds whose spines distinguish 4-manifold boundaries and thereby yield counterexamples to the deformation conjecture? The work further clarifies how spine existence behaves across dimensions, showing a sharp distinction at k=2 and providing a framework to explore the interplay between PL topology, handle theory, and 4-manifold invariants.

Abstract

We construct a compact PL 5-manifold $M$ (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, $\vee^{}_{1 1}S^2$, which is "spineless", meaning $M$ is not the regular neighborhood of any 2-complex PL embedded in $M$. We formulate a related question about the existence of exotic smooth structures on 4-manifolds which is of interest in relation to the deformation conjecture for 2-complexes, also known as the generalized Andrews-Curtis conjecture.

Theorems & Definitions (10)

• Theorem 1
• Remark 1.1
• Remark 1.2
• Theorem 2
• Proposition 1.3
• Theorem 3
• Proposition 2.1
• Lemma 2.2
• proof : Proof of Proposition \ref{['prop:vanishing']}
• proof