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 [math]\displaystyle{ \kappa(M) \in H^4(M;\mathbb{Z}/2) }[/math] 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 [math]\displaystyle{ TOP/PL \sim K(\mathbb Z/2,3) }[/math] 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 [math]\displaystyle{ E_8 \times T^n, n \geq 1 }[/math], where [math]\displaystyle{ E_8 }[/math] 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
References
- ↑ Kirby, Robion C.; Siebenmann, Laurence C. (1977). Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton, NJ: Princeton Univ. Pr.. ISBN 0-691-08191-3. http://www.maths.ed.ac.uk/~aar/papers/ks.pdf.
- ↑ Yuli B. Rudyak (2001). Piecewise linear structures on topological manifolds. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016.
- ↑ Francesco Polizzi. "Example of a triangulable topological manifold which does not admit a PL structure (answer on Mathoverflow)". https://mathoverflow.net/questions/214443/example-of-a-triangulable-topological-manifold-which-does-not-admit-a-pl-structu.
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