  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.
