Kirby–Siebenmann 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.

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.

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