9
$\begingroup$

I am having trouble coming up with an example of spaces where there exists a weak homotopy equivalence in one direction but not the other. Any hints or references are greatly appreciated!

Note: This is an instance of stagnating autodidactic studying, hence no home-work tag.

1 Answers 1

8

The circle $S^1$ is weakly equivalent to the so-called pseudocircle $\mathbb{S}$ (see wikipedia), and the weak equivalence goes $S^1 \to \mathbb{S}$. Any map $\mathbb{S}\to S^1$ induces the trivial map on $\pi_1$.

There are many more examples:

As shown by McCord (Singular homology groups and homotopy groups of finite topological spaces), any finite simplicial complex is weakly equivalent to a finite topological space.

  • 0
    This is a cool answer!2011-11-08