nLab
fundamental class

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

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

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Revised on October 1, 2014 02:50:27 by Anonymous Coward (128.197.60.253)