Spineless 5-manifolds and the deformation conjecture

Abstract

We construct a compact PL 5-manifold $M$ (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, ${\vee }_{11}^{}{S}^2$, 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.

7 pages. Version 2 is a substantial revision: the main theorem is strengthened, and an open question is formulated about the existence of exotic smooth structures on certain 4-manifolds. Version 3: improved exposition, updated references