Pro-algebraic homotopy types

Details Pro-algebraic homotopy types Pro-algebraic homotopy types

2006-06-05

arxiv.org/abs/math/0606107v8

Proc. London Math. Soc. 2008 97: 273-338

Mathematics

Algebraic Topology

72 pages; v6 Ref added to Baues-Lemaire conjecture (Rk 4.42); v7 minor changes, Sect. 5.1 corrected; v8 final version, to appe

Scientific paper

10.1112/plms/pdn004

The purpose of this paper is to generalise Sullivan's rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toen's schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Maurer-Cartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the Baues-Lemaire conjecture. For compact Kaehler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank, the schematic homotopy groups are shown to be \pi_n(X)\otimes k.

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