We construct a compact PL 5-manifold M (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, V₁₁ S²,
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.