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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically in
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a ''PL''-structure.


The KS-class

For a
topological manifold In topology, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds ...
''M'', the Kirby–Siebenmann class \kappa(M) \in H^4(M;\mathbb/2) is an element of the fourth
cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
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 Z/2,3) of ''TOP/PL'' with an
Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
.. 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 \geq 1, where E_8 stands for Freedman's
E8 manifold In low-dimensional topology, a branch of mathematics, the ''E''8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the ''E''8 lattice. History The E_8 manifold was discovered by Michael Freedman ...
. The class is named after
Robion Kirby Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he developed the Kirby–Siebenmann invariant ...
and Larry Siebenmann, who developed the theory of topological and ''PL''-manifolds.


See also

*
Hauptvermutung The ''Hauptvermutung'' of geometric topology is a now refuted conjecture asking whether any two Triangulation (topology), triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulati ...
*
Kervaire–Milnor group In mathematics, especially differential topology and cobordism theory, a Kervaire–Milnor group is an abelian group defined as the h-cobordism classes of homotopy spheres with the connected sum as composition and the reverse orientation as invers ...
, further obstruction for the existence of smooth structures


References

Homology theory Geometric topology Structures on manifolds Surgery theory {{topology-stub