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weak homotopy equivalence (Definition)

A continuous map $ f:X\rightarrow Y$ between path-connected based topological spaces is said to be a weak homotopy equivalence if for each $ k\ge 1$ it induces an isomorphism $ f_*:\pi_k(X)\rightarrow \pi_k(Y)$ between the $ k$th homotopy groups. $ X$ and $ Y$ are then said to be weakly homotopy equivalent.

Remark 1   It is not enough for $ \pi_k(X)$ to be isomorphic to $ \pi_k(Y)$ for all $ k.$ The definition requires these isomorphisms to be induced by a space-level map $ f.$



"weak homotopy equivalence" is owned by antonio.
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See Also: homotopy equivalence

Other names:  weak equivalence
Also defines:  weakly homotopy equivalent, weakly equivalent
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Cross-references: map, isomorphic, homotopy groups, isomorphism, induces, based topological spaces, path-connected, continuous map
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This is version 4 of weak homotopy equivalence, born on 2003-02-07, modified 2003-02-07.
Object id is 3987, canonical name is WeakHomotopyEquivalence.
Accessed 5275 times total.

Classification:
AMS MSC: 55P10 (Algebraic topology :: Homotopy theory :: Homotopy equivalences)
Pending Errata and Addenda
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