Kirby–Siebenmann class

Summary

The Kirby–Siebenmann class of a topological manifold $M$ is a cohomology class $\operatorname{ks}(M) \in H^4(M; \Z/2)$. It is the obstruction to putting a piecewise linear structure on $M \times \R$, and, when $\dim M \ge 5$, on $M$ itself. It shows that topological manifolds and piecewise linear manifolds really differ. In dimension $4$, a manifold whose class is nonzero, such as the $E_8$ manifold, has no smooth structure.