Kirby–Siebenmann class

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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[edit]

For a topological manifold M, the Kirby–Siebenmann 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 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 , 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.

See also[edit]


  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. Piecewise linear structures on topological manifolds. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016.
  3. ^ Francesco Polizzi. "Example of a triangulable topological manifold which does not admit a PL structure (answer on Mathoverflow)".