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weak homotopy equivalence
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(Definition)
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A continuous map $\funcdef{f}{X}{Y}$ between path-connected based topological spaces is said to be a weak homotopy equivalence if for each $k\ge 1$ it induces an isomorphism $\funcdef{f_*}{\pi_k(X)}{\pi_k(Y)}$ 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 $f.$
Remark 2 More generally, two spaces $X$ and $Y$ are defined to be weakly homotopy equivalent if there is a sequence of spaces and maps $$ X \to X_1 \from X_2 \to X_3 \from \cdots \to X_n \from Y$$ in which each map is a weak homotopy equivalence.
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"weak homotopy equivalence" is owned by antonio.
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Cross-references: sequence, map, isomorphic, homotopy groups, isomorphism, induces, based topological spaces, path-connected, continuous map
There are 9 references to this entry.
This is version 6 of weak homotopy equivalence, born on 2003-02-07, modified 2008-10-06.
Object id is 3987, canonical name is WeakHomotopyEquivalence.
Accessed 7312 times total.
Classification:
| AMS MSC: | 55P10 (Algebraic topology :: Homotopy theory :: Homotopy equivalences) |
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Pending Errata and Addenda
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