# 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 [1] and is stated as follows.

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

[1]

## References

1. Spanier, E. H.: 1966, Algebraic Topology , McGraw Hill: New York.