Hopf–Whitney theorem

For a n-dimensional CW complex X and a n-1-connected space Y, the well-defined map: : [X,Y]\rightarrow H^n(X,\pi_n(Y)), [f]\mapsto f^*\iota with a certain cohomology class \iota\in H^n(Y,\pi_n(Y)) is an isomorphism. The Hurewicz theorem claims that the well-defined map \pi_n(Y)\rightarrow H_n(Y,\mathbb{Z}),[f]\mapsto f_*[S^n] with a fundamental class [S^n]\in H_n(S^n,\mathbb{Z})\cong\mathbb{Z} is an isomorphism and that H_{n-1}(Y,\mathbb{Z})\cong 1, which implies \operatorname{Ext}_\mathbb{Z}^1(H_{n-1}(Y,\mathbb{Z}),\pi_n(Y))\cong 1 for the Ext functor. The Universal coefficient theorem then simplifies and claims: : H^n(Y,\pi_n(Y)) \cong\operatorname{Hom}_\mathbb{Z}(H_n(Y,\mathbb{Z}),\pi_n(Y)) \cong\operatorname{End}_\mathbb{Z}(\pi_n(Y)). \iota\in H^n(Y,\pi_n(Y)) is then the cohomology class corresponding to the identity \operatorname{id} \in\operatorname{End}_\mathbb{Z}(\pi_n(Y)). In the Postnikov tower removing homotopy groups from above, the space Y_n only has a single nontrivial homotopy group \pi_n(Y_n)\cong\pi_n(Y) and hence is an Eilenberg–MacLane space K(\pi_n(Y),n) (up to weak homotopy equivalence), which classifies singular cohomology. Combined with the canonical map Y\rightarrow Y_n\simeq K(\pi_n(Y),n), the map from the Hopf–Whitney theorem can alternatively be expressed as a postcomposition: : [X,Y]\rightarrow[X,K(\pi_n(Y),n)]\cong H^n(X,\pi_n(Y)). == Examples ==

For homotopy groups, cohomotopy sets or cohomology, the Hopf–Whitney theorem reproduces known results but weaker: • For every n-1-connected space Y one has: : [S^n,Y] \cong H^n(S^n,\pi_n(Y)) \cong\pi_n(Y). : In general, this holds for every topological space by definition. • For a n-dimensional CW complex X one has: : [X,S^n] \cong H^n(X,\pi_n(S^n)) \cong H^n(X,\mathbb{Z}). : For n=1, this also follows from S^1\simeq K(\mathbb{Z},1). • For a topological group G and a natural number n, the Eilenberg–MacLane space K(G,n) is n-1-connected by construction, hence for every n-1-dimensional CW-complex X one has: : [X,K(G,n)] \cong H^n\left(X,\pi_nK(G,n)\right) \cong H^n(X,G) : In general, this holds for every topological space. The Hopf–Whitney theorem produces a weaker result because the fact that the higher homotopy groups of an Eilenberg–MacLane space also vanish does not enter. == Literature ==