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generalized Hurewicz fundamental theorem
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(Theorem)
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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.
- 1
- Spanier, E. H.: 1966, Algebraic Topology, McGraw Hill: New York.
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"generalized Hurewicz fundamental theorem" is owned by bci1.
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See Also: CW complex
| Other names: |
general Hurewicz Theorem |
| Also defines: |
extended Hurewicz Fundamental Theorem |
| Keywords: |
generalization of the Hurewicz Fundamental Theorem |
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Cross-references: isomorphism, homology groups, homotopy groups, topological spaces, CW-complexes, connected, theorem
This is version 10 of generalized Hurewicz fundamental theorem, born on 2008-07-19, modified 2009-06-10.
Object id is 10834, canonical name is GeneralizedHurewiczFundamentalTheorem.
Accessed 998 times total.
Classification:
| AMS MSC: | 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces ) | | | 57Q05 (Manifolds and cell complexes :: PL-topology :: General topology of complexes) | | | 54A05 (General topology :: Generalities :: Topological spaces and generalizations ) | | | 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces ) | | | 57Q12 (Manifolds and cell complexes :: PL-topology :: Wall finiteness obstruction for CW-complexes) | | | 57N60 (Manifolds and cell complexes :: Topological manifolds :: Cellularity) | | | 55U10 (Algebraic topology :: Applied homological algebra and category theory :: Simplicial sets and complexes) | | | 18G30 (Category theory; homological algebra :: Homological algebra :: Simplicial sets, simplicial objects ) |
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Pending Errata and Addenda
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