I read in several places that "Every $Y^3$ with the integral homology of $S^3$ bounds a contractible $4$-manifold $\Delta^4$".
But in Kervaire's paper "Smooth homology spheres and their fundamental groups" it was written "The following well-known construction provides an example of a 3-dimensional homology sphere which does not bound a contractible manifold."
What did I miss?
If it was a suitable for MO, I will move it to SE.
Best Answer
If Poincare homology sphere(with negative orientation) bounds a smooth contractible 4 manifold then cap it off with an $-E_8$ plumbing will give you a simplyconnected 4 manifold with $-E_8$ intersection form [It's a -ve definite 4 manifold]. The famous theorem of Donaldson says that if a closed , simplyconnected 4 manfold is -ve definite then the intersection form has to be diagonilazable. Although $E_8$ is not diagonalizable. So contradiction.
Although Freedman's famous result says every homology sphere embedded inside $S^4$ in a topological sense. That means it bounds a contractible 4 manifold in topological category.