group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a connected oriented closed manifold, its integral homology group in degree the dimension of $X$ is isomorphic to the integers
The generator of this corresponding to the choice of orientation is called the fundamental class of $X$.
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.
(…)
Last revised on June 17, 2015 at 13:16:47. See the history of this page for a list of all contributions to it.