Weak homotopy equivalence of topological spaces

Definition for path-connected spaces in terms of homotopy groups
Let $$A$$ and $$B$$ be path-connected spaces. A weak homotopy equivalence from $$A$$ to $$B$$ is a continuous map $$f:A \to B$$ such that the functorially induced maps $$\pi_n(f):\pi_n(A) \to \pi_n(B)$$ are group isomorphisms for all $$n \ge 1$$.

Note that since the maps are homomorphisms anyway, it is enough to require them to be bijective.

Basepoint choice disclaimer for homotopy group isomorphism: To concretely define the map $$\pi_n(f)$$, we need to choose basepoints for $$A$$ and $$B$$. Change of basepoint, however, only results in pre-composition and post-composition by isomorphisms and does not affect whether or not the map is an isomorphism.

Equivalent definition for path-connected spaces in terms of homology groups
Let $$A$$ and $$B$$ be path-connected spaces. Then a continuous map $$f:A \to B$$ is a weak homotopy equivalence iff both of these conditions hold:


 * The induced map $$\pi_1(f):\pi_1(A) \to \pi_1(B)$$ is an isomorphism of groups.
 * For every bundle of abelian groups $$\mathcal A$$ over $$B$$, the induced map of twisted homology groups $$f_*:H_n(A;f^* \mathcal A) \to H_n(B;\mathcal A)$$ is an isomorphism of groups for all $$n \ge 0$$.

As above, all of these maps are homomorphisms anyway, so it is enough to require them to be bijective. The above basepoint disclaimer for $$\pi_1$$ also applies here.

If both $$A$$ and $$B$$ are simply connected then the criterion is simpler: a continuous map $$f:A \to B$$ is a weak homotopy equivalence iff the induced map on homology with $$\mathbb Z$$ coefficients $$f_*:H_n(A; \mathbb Z) \to H_n(B; \mathbb Z)$$ is an isomorphism of groups for all $$n \ge 0$$.

Definition for spaces that are not path-connected
Let $$A$$ and $$B$$ be topological spaces. A weak homotopy equivalence from $$A$$ to $$B$$ is a continuous map $$f:A \to B$$ such that:


 * The functorially induced map $$\pi_0(f): \pi_0(A) \to \pi_0(B)$$ is a bijection between the set of path components $$\pi_0(A)$$ and the set of path components $$\pi_0(B)$$.
 * For every path component of $$A$$, the restriction of $$f$$ to a continuous map from that to its image path component of $$B$$ is a weak homotopy equivalence of path-connected spaces.

Facts

 * The existence of a weak homotopy equivalence from $$A$$ to $$B$$ does not imply the existence of a weak homotopy equivalence from $$B$$ to $$A$$. Thus, to get an equivalence relation on topological spaces, we need to take a symmetric transitive closure. We say that spaces are weak homotopy-equivalent topological spaces if they are in the same equivalence class under the equivalence relation thus obtained.
 * The mere fact that $$\pi_n(A) \cong \pi_n(B)$$ as abstract groups is not enough to guarantee that $$A$$ and $$B$$ are weak homotopy-equivalent, even when $$A$$ and $$B$$ are manifolds or CW-spaces (see isomorphic homotopy groups not implies weak homotopy-equivalent). Rather, it is specifically important that the map must induce those isomorphisms.
 * The exception to the above is in the case that both $$\pi_n(A)$$ and $$\pi_n(B)$$ are the trivial group/one-point set for all $$n$$. In this case, any map must induce isomorphisms since that's the only possible map between trivial groups/one-point sets. In this case, the spaces $$A$$ and $$B$$ are both weakly contractible spaces.
 * Similarly, the mere fact that $$\pi_1(A) \cong \pi_1(B)$$ as abstract groups and $$H_n(A) \cong H_n(B)$$ as abstract groups does not imply that $$A$$ and $$B$$ are weak homotopy-equivalent. See isomorphic homology groups and isomorphic fundamental groups not implies weak homotopy-equivalent. Rather, it is specifically important that the map must induce those isomorphisms.
 * The exception to the above is, once again, where the fundamental group and all the homology groups $$H_n, n \ge 1$$, are trivial.

Stronger properties

 * Homotopy equivalence of topological spaces

Weaker properties

 * Homology isomorphism of topological spaces