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# generalized Hurewicz fundamental theorem

# 1 Generalized Hurewicz fundamental theorem

The Hurewicz theorem was generalized from connected CW-complexes to arbitrary topological spaces [1] and is stated as follows.

###### Theorem 1.1.

(Generalized Hurewicz Fundamental 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 isomorphism.*

# References

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

Defines:

extended Hurewicz Fundamental Theorem

Keywords:

generalization of the Hurewicz Fundamental Theorem

Related:

CWComplex

Synonym:

general Hurewicz Theorem

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

18G30*no label found*55U10

*no label found*57N60

*no label found*57Q12

*no label found*54D05

*no label found*54A05

*no label found*57Q05

*no label found*54D05

*no label found*

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