PlanetPhysics/Generalized Hurewicz Fundamental Theorem

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Generalized Hurewicz Fundamental Theorem

The Hurewicz theorem was generalized from connected CW-complexes to arbitrary topological spaces and is stated as follows.

\begin{theorem} If $\pi _{r}(K,L)=0$ for $1\leq r\leq n$ , $(n\geq 2)$, then $h_{\pi }:\pi _{n}^{*}(K,L)\simeq H_{n}(K,L)$ , where $\pi _{n}$ are homotopy groups, $H_{n}$ are homology groups, K and L are arbitrary topological spaces, and `$\simeq$' denotes an \htmladdnormallink{isomorphism {http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}.} \end{theorem}