In mathematics, more specifically in geometric topology, the KirbyâSiebenmann class is an obstruction for topological manifolds to allow a PL-structure. [1]
For a topological manifold M, the KirbyâSiebenmann class 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 of TOP/PL with an EilenbergâMacLane space.
The Kirby-Siebenmann class can be used to prove the existence of topological manifolds that do not admit a PL-structure. [2] Concrete examples of such manifolds are , where stands for Freedman's E8 manifold. [3]
The class is named after Robion Kirby and Larry Siebenmann, who developed the theory of topological and PL-manifolds.