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homotopy equivalence
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(Definition)
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Definition Suppose that  and  are topological spaces and
 is a continuous map. If there exists a continuous map  such that
 (i.e.  is homotopic to the identity mapping on  ), and
 , then  is a homotopy equivalence. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence.
If there exist a homotopy equivalence between the topological spaces  and  , we say that  and  are homotopy equivalent, or that  and  are of the same homotopy type. We then write  .
- Any homeomorphism
 is obviously a homotopy equivalence with  .
- For topological spaces, homotopy equivalence is an equivalence relation.
- A topological space
 is (by definition) contractible, if  is homotopy equivalent to a point, i.e.,
 .
- 1
- A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Also available online.
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"homotopy equivalence" is owned by matte. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: point, contractible, equivalence relation, homeomorphism, weak homotopy equivalence, identity mapping, continuous map, topological spaces
There are 27 references to this entry.
This is version 10 of homotopy equivalence, born on 2002-01-23, modified 2003-07-10.
Object id is 1589, canonical name is HomotopyEquivalence.
Accessed 9594 times total.
Classification:
| AMS MSC: | 55P10 (Algebraic topology :: Homotopy theory :: Homotopy equivalences) |
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Pending Errata and Addenda
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