nLab mod p Whitehead theorem

Idea

A basic fact of p-adic homotopy theory, roughly analogous to the plain Whitehead theorem in plain homotopy theory.

Let $p$ be a prime number and write $\mathbb{F}_p \coloneqq \mathbb{Z}/(p)$ for its cyclic group/finite field and $\mathbb{Z}_p$ for the p-adic integers.

Statement For $X$ and $Y$ simply connected finite homotopy types and a morphism $f \colon X \longrightarrow Y$ between them, the following are equivalent:

1. $f$ is an $\mathbb{F}_p$-homology equivalence, i.e. $H_\ast(f,\mathbb{F}_p)$ is an equivalence;

2. $f$ is a $\mathbb{Z}_p$-homology equivalence, i.e. $H_\ast(f,\mathbb{Z}_p)$ is an equivalence;

3. $f$ is a $\mathbb{Z}_p$-weak homotopy equivalence, i.e. $\pi_\ast(f) \otimes \mathbb{Z}_p$ is an equivalence.

(Schiffmann 81)

Applications

This theorem controls the relation between localization of a space at $\mathbb{F}_p$ and its p-completion.

References

• Stephen J. Schiffman, A mod p Whitehead Theorem, Proceedings of the American Mathematical Society Vol. 82, No. 1 (May, 1981), pp. 139-144