In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.
The KS-class
For a topological manifold M, the Kirby–Siebenmann class <math>\kappa(M) \in H^4(M;\mathbb{Z}/2)</math> is an element of the fourth cohomology group of M that vanishes if M admits a piecewise linear structure.
It is the only such obstruction, which can be phrased as the weak equivalence <math> TOP/PL \sim K(\mathbb Z/2,3) </math> of TOP/PL with an Eilenberg–MacLane space..
The Kirby-Siebenmann class can be used to prove the existence of topological manifolds that do not admit a PL-structure. Concrete examples of such manifolds are <math> E_8 \times T^n, n \geq 1</math>, where <math> E_8 </math> stands for Freedman's E8 manifold.
The class is named after Robion Kirby and Larry Siebenmann, who developed the theory of topological and PL-manifolds.
See also
- Hauptvermutung
- Kervaire–Milnor group, further obstruction for the existence of smooth structures