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

Brown, Edgar H.; Szczarba, Robert H.

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.

Reviewer: H.M.Unsöld (Berlin)

MSC:

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

Keywords:

minimal algebras; cochain algebra; Lie algebra; simplicial realization; simplicial space; real homotopy type; Lie group; simplicial fibration; local coefficients; de Rham theorem; characteristic classes; transverse G-foliations; Serre spectral sequence; continuous de Rham cohomology; continuous cohomology

Citations:

Zbl 0721.00021; Zbl 0671.55006; Zbl 0374.57002; Zbl 0387.57013