Kirby–Siebenmann class

Remarks

The $4$-dimensional $E_8$ manifold $M_{E_8}$ has $\operatorname{ks}(M_{E_8}) \ne 0$ in $H^4(M_{E_8}; \Z/2) \cong \Z/2$. For a closed simply connected topological $4$-manifold with even intersection form, $\operatorname{ks}$ is the signature divided by $8$, taken modulo $2$, and the $E_8$ form has signature $8$.

A space that is not a manifold, such as a finite graph with a vertex of degree three, has no tangent microbundle, so the definition does not apply to it.

If $M$ has a piecewise linear structure, or a smooth structure, then $\operatorname{ks}(M) = 0$. Conversely, if $\dim M \ge 5$ and $\operatorname{ks}(M) = 0$, then $M$ has a piecewise linear structure; for any $M$, $\operatorname{ks}(M) = 0$ if and only if $M \times \R$ has one. The class is natural: for an open subset $U \subseteq M$, $\operatorname{ks}(U)$ is the restriction of $\operatorname{ks}(M)$.