nLab rational homotopy equivalence

Idea

A rational homotopy equivalence is the notion of equivalence of topological spaces as used in rational homotopy theory. Where a weak homotopy equivalence in ordinary homotopy theory identifies spaces under morphisms that induce isomorphisms on all homotopy groups, rational homotopy equivalences identify spaces under morphisms that induce isomorphisms on all rationalized homotopy groups.

Definition

Definition

For $X$ and $Y$ be simply connected topological spaces and $f : X \to Y$ a continuous map between them, $f$ is called a rational homotopy equivalence if the following equivalent conditions are satisfied:

1. it induces an isomorphism on rationalized homotopy groups: $\pi_*(f) \otimes \mathbb{Q} : \pi_*(X) \otimes \mathbb{Q} \stackrel{\simeq}{\to} \pi_*(X) \otimes \mathbb{Q}$;

2. it induces an isomorphism on ordinary homology groups with rational coefficients: $H_*(f,\mathbb{Q}) : H_*(X,\mathbb{Q}) \stackrel{\simeq}{\to} H_*(X,\mathbb{Q})$;

3. it induces an isomorphism on rational cohomology groups: $H^*(f,\mathbb{Q}) : H^*(X,\mathbb{Q}) \stackrel{\simeq}{\to} H^*(X,\mathbb{Q})$;

4. it induces a weak homotopy equivalence on the rationalizations $X_{ra}$ and $Y_{ra}$ : $f_{ra} : X_{ra} \stackrel{\simeq_{whe}}{\to} Y_{ra}$.

That (the first two of) these conditions are equivalent is due to Serre 53

References

The concept is due to

• Jean-Pierre Serre, Groupes d’homotopy et classes de groupes abelians, Ann. of Math. 58 (1953) 258-294

Review includes

• Kathryn Hess, around definition 1.17 in Rational homotopy theory: a brief introduction (arXiv:math.AT/0604626)