---
tag: E4NJER1RFQ
kind: concept
name: Kirby–Siebenmann class
msc: [57Q25, 57N55]
sources:
  - url: https://en.wikipedia.org/wiki/Kirby%E2%80%93Siebenmann_class
same_as:
  wikidata: Q6414983
  wikipedia: 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.

## Explanations

### L1 human

Some topological manifolds can be cut into simplices in a way that makes them piecewise linear, and some cannot.
Kirby and Siebenmann found a single test for this, in the form of one cohomology class with coefficients in $\Z/2$.
If the class is nonzero, no piecewise linear or smooth structure exists.
In dimensions five and higher, a zero class means that a piecewise linear structure does exist.