Rational homotopy theory for non-simply connected spaces

Abstract

Abstract. We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.The rationalisation of an abelian group Γ is the rational vector space Γ ⊗ Q, together with the obvious homomorphism Γ → Γ ⊗ Q. Rational homotopy theory begins with the introduction by Sullivan [20] of a geometric analogue: with any simply connected CW space X there is associated a continuous map f : X → X Q with X Q simply connected, and such that π i (X Q ) = π i (X) ⊗ Q and π i (f ) is the rationalisation. The homotopy type of X Q is called the rational homotopy type of X, the space X Q is called the rationalisation of X, and X is called a rational space if the groups π i (X) themselves are rational -in which case f is a homotopy equivalence.A principal feature of rational homotopy theory in the simply connected case, as developed by Quillen [18] and Sullivan [21], is that the geometric construction X Q can be replaced by an equivalent algebraic construction. Sullivan's approach consists of three steps. First, a simplicial commutative cochain algebra {A P L (n), ∂ i , s i } is used to define a functor A P L (−) from connected simplicial sets (and topological spaces) to the category A of commutative cochain algebras A satisfying H 0 (A) = Q. Second, a distinguished class of these cochain algebras is introduced, now called minimal Sullivan algebras. Any cochain algebra A ∈ A admits a quasi-isomorphism M → A from a uniquely determined minimal Sullivan algebra, the minimal Sullivan model of A. If X is a connected topological space, then the minimal model M X of A P L (X) is a Sullivan model for X. Finally within A the full subcategory M of minimal Sullivan algebras is equipped with a notion of homotopy. A homotopy preserving functor is then constructed from M to

Schematic homotopy types and non-abelian Hodge theory

“…As explained before, our construction of the Hodge decomposition uses equivariant co-simplicial algebras as algebraic models for schematic homotopy types. These algebraic models are very close to the equivariant differential graded algebras of [BS93,GHT99]. The main difference between the two approaches is that we use co-simplicial algebras equipped with an action of an affine group scheme, whereas in [BS93,GHT99] the authors use algebras equivariant for a discrete group action.…”

“…These algebraic models are very close to the equivariant differential graded algebras of [BS93,GHT99]. The main difference between the two approaches is that we use co-simplicial algebras equipped with an action of an affine group scheme, whereas in [BS93,GHT99] the authors use algebras equivariant for a discrete group action. In a sense, our approach is an algebraization of their approach, adapted for the purpose of Hodge theory.…”

An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups

“…Gómez-Tato, Halperin and Tanré developed a version of rationalizations of spaces that I call it π 1 -fiberwise rationalization [15]. They also developed a theory of algebraic models that extends the theory of minimal Sullivan modules.…”

“…(3) Casacuberta-Peschke's Ω-rationalization L ΩQ [12]; (4) Gómez-Tato-Halperin-Tanré's π 1 -fiberwise rationalization L π 1 Q [15]. All these functors are non-isomorphic but there are natural transformations:…”

Formality of $k$-connected spaces in $4k+3$ and $4k+4$ dimensions

“…If the manifold has a nontrivial fundamental group, the proof of Theorem 2 will not work and the alternative definition given by Proposition 1 has to be adapted. Different ways to introduce the fundamental group in this picture were studied in [12] and led to nonequivalent definitions of formality.…”