Wall's finiteness obstruction

Summary

Wall's finiteness obstruction is an element $\sigma(X)$ attached to a finitely dominated CW complex $X$. It lies in the reduced K0 of the integral group ring of $\pi_1(X)$. It vanishes exactly when $X$ is homotopy equivalent to a finite CW complex. It is a basic example of how algebraic K-theory measures obstructions in topology.