Kirby–Siebenmann class

In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.^[1]

The KS-class

For a topological manifold M, the Kirby–Siebenmann class ${\displaystyle \kappa (M)\in H^{4}(M;\mathbb {Z} /2)}$ 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 ${\displaystyle TOP/PL\sim K(\mathbb {Z} /2,3)}$ of TOP/PL with an Eilenberg–MacLane space.^{[clarification needed]}.

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 ${\displaystyle E_{8}\times T^{n},n\geq 1}$, where ${\displaystyle E_{8}}$ 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.

See also

• Hauptvermutung
• Kervaire–Milnor group, further obstruction for the existence of smooth structures