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.

The KS-class

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

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 $E_8\times T^n,n\ge 1$, where $E_8$ 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

References

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