Kirby–Siebenmann class

For a topological manifold M, the Kirby–Siebenmann class $\backslash kappa(M)\; \backslash in\; H^4(M;\backslash mathbb\{Z\}/2)$ 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 $TOP/PL\; \backslash sim\; K(\backslash mathbb\; Z/2,3)$ of TOP/PL with an Eilenberg–MacLane space.{{clarify|reason=The clause beginning with "which" is preceded by a comma, creating ambiguity. It is unclear whether the clause is meant to restrict "obstruction" (indicating a unique characterization) or to provide additional, nonessential commentary. Please revise the sentence for clarity by either removing the comma or rephrasing the clause so that its intended relationship to the rest of the sentence is unmistakable.|date=February 2025}}.

The Kirby-Siebenmann class can be used to prove the existence of topological manifolds that do not admit a PL-structure.{{cite book|title=Piecewise linear structures on topological manifolds|author=Yuli B. Rudyak|year=2001|publisher=World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016|arxiv=math/0105047}} Concrete examples of such manifolds are $E\_8\; \backslash times\; T^n,\; n\; \backslash geq\; 1$, where $E\_8$ stands for Freedman's E8 manifold.{{cite news|author=Francesco Polizzi|title=Example of a triangulable topological manifold which does not admit a PL structure (answer on Mathoverflow)|url=https://mathoverflow.net/questions/214443/example-of-a-triangulable-topological-manifold-which-does-not-admit-a-pl-structu}}

The class is named after Robion Kirby and Larry Siebenmann, who developed the theory of topological and PL-manifolds.

• Hauptvermutung
• Kervaire–Milnor group, further obstruction for the existence of smooth structures

{{Reflist}}

{{DEFAULTSORT:Kirby-Siebenmann class}}

Category:Homology theory

Category:Geometric topology

Category:Structures on manifolds

Category:Surgery theory

{{topology-stub}}