nLab p-adic homotopy theory

Idea

In $p$-adic homotopy theory one studies, for any prime number $p$, 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 $\mathbb{Z}_p$ – the $p$-adic homotopy types. The central theorem (Mandell 01) says that the (∞,1)-category on $p$-adic homotopy types is faithfully embedded into the (∞,1)-category of E-∞ algebras over an algebraically closed field of characteristic $p$ (Lurie, p. 70, cor. 3.5.15).

In this way $p$-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 $p$-adic homotopy theory for each prime $p$.

Related concepts

• p-adic numbers

• p-completion, p-localization

• mod p Whitehead theorem

• p-adic geometry

• p-adic cohomology

• p-adic physics

• rational homotopy theory, real homotopy theory

References

• Michael Mandell, $E_\infty$-algebras and $p$-Adic homotopy theory, Topology 40 (2001), no. 1, 43-94. (pdf)

  surveyed in: Algebraic models in $p$-adic homotopy theory, YTM13, 2013 (pdf)

• Jacob Lurie, $p$-Adic homotopy theory (pdf)

• Jacob Lurie, Rational and p-adic Homotopy Theory