nLab fundamental class





Special and general types

Special notions


Extra structure





For manifolds

For XX a connected oriented closed manifold, its integral homology group in degree the dimension of XX is isomorphic to the integers

H dimX(X,). H_{dim X}(X, \mathbb{Z}) \simeq \mathbb{Z} \,.

The generator of this corresponding to the choice of orientation is called the fundamental class of XX.

For (n1)(n-1)-connected spaces

If a topological space XX is (n1)(n-1)-connected for n2,n\geq2, then by the Hurewicz theorem there is an isomorphism h:π n(X)H n(X)h\colon\pi_n(X)\to H_n(X). By the universal coefficient theorem, we have H n(X;π n(X))=hom(H n(X),π n(X))H^n(X;\pi_n(X))=\hom(H_n(X),\pi_n(X)). Hence h 1h^{-1} represents an element of H n(X;π n(X))H^n(X;\pi_n(X)) called the fundamental class of XX. In particular, the Eilenberg-MacLane space K(G,n)K(G,n) has a fundamental class ι\iota which represents the identity map 1[K(G,n),K(G,n)]H n(K(G,n);G).1\in [K(G,n),K(G,n)]\cong H^n(K(G,n);G). This is the universal cohomology class, in the sense that all cohomology classes are pullbacks of this one by classifying maps. ref Mosher and Tangora.

Virtual fundamental class


Last revised on June 17, 2015 at 17:16:47. See the history of this page for a list of all contributions to it.