Continuous cohomology and real homotopy type. II. (English) Zbl 0731.55008

Théorie de l’homotopie, Colloq. CNRS-NSF-SMF, Luminy/Fr. 1988, Astérisque 191, 45-70 (1990).
[For the entire collection see Zbl 0721.00021.]
[For the review of the first part see Trans. Am. Math. Soc. 311, No.1, 57-106 (1989; Zbl 0671.55006).]
The authors study minimal algebras in the sense of D. Sullivan [Publ. Math. Inst. Haut. Étud. Sci. 47, 269-331 (1977; Zbl 0374.57002)], which are not necessarily nilpotent. The main result is a proof of “Theorem” 8.1. in the cited paper. Let (A,d) be a minimal algebra of finite type over the reals and let L be the dual of \(A^ 1\). The dual of d induces a Lie bracket on L, and \(A^{(1)}\), the subalgebra generated by \(A^ 1\), is isomorphic to the cochain algebra \(C^*(L)\) of the Lie algebra L. Let \(\Delta\) A be the simplicial realization of A. This is a simplicial space which represents the real homotopy type determined by A, see the first part of the paper.
(i) If G is the simply connected Lie group with Lie algebra L, then \(\pi_ 1(\Delta A)\approx G\) and \(\pi_ q(\Delta A)\approx \pi_ q(G)\) (q\(\geq 2).\)
(ii) If \(A'=A/(ideal\) generated by \(A^ 1)\) then \(A^{(1)}\subset A\) induces a simplicial fibration \(\Delta A\to \Delta A^{(1)}\) with fibre \(\Delta A'\). Since \(A'\) is simply connected, the methods of part I of the paper apply to \(\Delta A'.\)
(iii) If L acts on a finite dimensional vector space V, then this action induces a system \(\{\) \(V\}\) of local coefficients on \(\Delta\) A and there is a differential d on \(A\otimes V\) such that the de Rham theorem with local coefficients holds, i.e. \(H^*(A\otimes V,d)\approx H^*(\Delta A;\{V\}).\)
As an application the authors prove a theorem on characteristic classes of G-foliations transverse to the fibres of a product [see A. Haefliger, Lect. Notes Math. 652, 1-12 (1978; Zbl 0387.57013)].
Let G be a compact Lie group with Lie algebra L and let \(BL=\Delta C^*(L)^{\delta}\) be the classifying space of transverse G-foliations. Then the natural map \(H^*(C^*(L))\to H^*(BL;{\mathbb{R}})\) is injective.
The technical part of the paper centres around simplicial fibrations and a Serre spectral sequence (with twisted coefficients) for continuous de Rham cohomology. The appendix contains a proof (due to G. Segal) of the fact that for CW-complexes cohomology with real coefficients and continuous cohomology agree.


55P60 Localization and completion in homotopy theory
55N25 Homology with local coefficients, equivariant cohomology
55U10 Simplicial sets and complexes in algebraic topology