No abstract

No abstract

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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

Given an N -dimensional compact closed oriented manifold M and a field lk, F. Cohen and L. Taylor have constructed a spectral sequence, E(M, n, lk), converging to the cohomology of the space of ordered configurations of n points in M . The symmetric group Σn acts on this spectral sequence giving a spectral sequence of Σn -differential graded commutative algebras. Here, an explicit description is provided of the invariants algebra (E 1 , d 1 ) Σn of the first term of E(M, n, Q). This determination is applied in two directions.(a) In the case of a complex projective manifold or of an odd-dimensional manifold M , the cohomology algebra H * (Cn (M ); Q) of the space of unordered configurations of n points in M is obtained (the concrete example of P 2 (C) is detailed).(b) The degeneration of the spectral sequence formed of the Σn -invariants E(M, n, Q) Σn at level 2 is proved for any manifold M .These results use a transfer map and are also true with coefficients in a finite field Fp with p > n.