Special and general types
For a connected oriented closed manifold, its integral homology group in degree the dimension of is isomorphic to the integers
The generator of this corresponding to the choice of orientation is called the fundamental class of .
For -connected spaces
If a topological space is -connected for then by the Hurewicz theorem there is an isomorphism . By the universal coefficient theorem, we have . Hence represents an element of called the fundamental class of . In particular, the Eilenberg-MacLane space has a fundamental class which represents the identity map 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