nLab fundamental class

Definition

For manifolds

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

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

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

For $(n-1)$-connected spaces

If a topological space $X$ is $(n-1)$-connected for $n\geq2,$ then by the Hurewicz theorem there is an isomorphism $h\colon\pi_n(X)\to H_n(X)$. By the universal coefficient theorem, we have $H^n(X;\pi_n(X))=\hom(H_n(X),\pi_n(X))$. Hence $h^{-1}$ represents an element of $H^n(X;\pi_n(X))$ called the fundamental class of $X$. In particular, the Eilenberg-MacLane space $K(G,n)$ has a fundamental class $\iota$ which represents the identity map $1\in [K(G,n),K(G,n)]\cong H^n(K(G,n);G).$ 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

(…)

• virtual fundamental class.

Related concepts

• Poincaré duality complex

• Poincaré duality algebra

• intersection theory

• wrapped brane