A spine of a manifold M is a piecewise-linear, lower-dimensional submanifold (not necessarily locally flat) whose embedding is a homotopy equivalence. The question of which manifolds admit spines has a rich history going back to the 1960s. For any m>4, Cappell and Shaneson showed in the 1970s that if an m-dimensional manifold M is homotopy equivalent to an (m-2)-dimensional submanifold N, and either m is odd or M is simply connected, then M contains N as a spine. In contrast, I will show that there exist smooth, compact, simply-connected 4-manifolds that are homotopy equivalent to the 2-sphere but do not contain a spine (joint work with Tye Lidman). I will also discuss some related results about piecewise-linear concordance of knots in homology spheres (joint with Lidman and Jen Hom).