nLab mod p Whitehead theorem



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

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

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

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

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

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

(Schiffmann 81)


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


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

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