Is there any way of proving two spaces have the same homotopy class without using directly the definition of homotopy equivalence?

Question

Given two spaces $X,Y$ homotopically equivalent sometimes it is not easy to find eplicit maps $f:X\to Y$ and $g:Y:\to X$ satisfying the definition of homotopic equivalence. I want to know if there are other ways of proving they are equivalent, probably adding some conditions on $X$ and $Y$ (like having CW-complex structure). For example, is there any situation in which having isomorphic fundamental group implies having same homotopy type?

Answer 1

There is a concept called “weak homotopy equivalence” which is defined in terms of algebraic invariants: two spaces $X$ and $Y$ are weakly homotopy equivalent if there is a map $X \to Y$ inducing isomorphisms on all higher homotopy groups.

It is known for certain classes of spaces—most notably CW complexes—that two spaces are weakly homotopy equivalent iff they are homotopy equivalent (Whitehead’s Theorem). However, I can’t think of a practical example where it is easier to apply Whitehead’s theorem than to construct a homotopy equivalence. I’ve mostly seen it used to prove other things.