nLab
p-adic homotopy theory

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

In pp-adic homotopy theory one studies, for any prime number pp, simply connected homotopy types (of topological spaces, hence ∞-groupoids) all of whose homotopy groups have the structure of (finitely generated) modules over the p-adic integers p\mathbb{Z}_p – the pp-adic homotopy types. The central theorem (Mandell 01) says that the (∞,1)-category on pp-adic homotopy types is faithfully embedded into the (∞,1)-category of E-∞ algebras over an algebraically closed field of characteristic pp (Lurie, p. 70, cor. 3.5.15).

In this way pp-adic homotopy theory is directly analogous to rational homotopy theory with the rational numbers \mathbb{Q} replaced by the p-adic integers.

The fracture theorem says that under mild conditions, homotopy theory may be decomposed in a precise sense into rational homotopy theory and pp-adic homotopy theory for each prime pp.

References

Revised on April 20, 2016 09:22:24 by Tim Porter (95.144.140.103)