The blob complex.
by Scott Morrison
Abstract: Topological field theories associate vector spaces to n-manifolds (and sometimes numbers to (n+1)-manifolds, although that won't concern us much). The blob complex provides a natural way to see these vector spaces as the 0-th homology of an invariant chain complex. I'll explain some nice properties of this chain complex, and tell you what we might hope to learn about manifolds by looking at the higher homologies of this chain complex.