PlanetPhysics/Generalized Hurewicz Fundamental Theorem

\newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}&

\eqno{\mbox{#9}}$$ }

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}