Homotopy class

Applications

Homotopy theory can be used as a foundation for homology theory: one can represent a cohomology functor on a space X by mappings of X into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian group G, and any based CW-complex X, the set ${\displaystyle [X,K(G,n)]}$ of based homotopy classes of based maps from X to the Eilenberg–MacLane space${\displaystyle K(G,n)}$ is in natural bijection with the n-th singular cohomology group ${\displaystyle H^{n}(X,G)}$ of the space X. One says that the omega-spectrum of Eilenberg-MacLane spaces are representing spaces for singular cohomology with coefficients in G. Using this fact, homotopy classes between a CW complex and a multiply connected space can be calculated using cohomology as described by the Hopf–Whitney theorem.