##
**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.

[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 |