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

References

1. Kirby, Robion C.; Siebenmann, Laurence C. (1977). Foundational Essays on Topological Manifolds, Smoothings, and Triangulations (PDF). Princeton, NJ: Princeton Univ. Pr. ISBN 0-691-08191-3.
2. Yuli B. Rudyak (2001). Piecewise linear structures on topological manifolds. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. arXiv:math/0105047.
3. Francesco Polizzi. "Example of a triangulable topological manifold which does not admit a PL structure (answer on Mathoverflow)".