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**Continuous cohomology and real homotopy type.**
*(English)*
Zbl 0671.55006

A homotopy theory of simplicial spaces is developed. It is shown that Kan’s definition of \(\pi_*\) generalizes to simplicial spaces and gives topological groups as homotopy groups. Moreover, the concepts of fibration and continuous cohomology are developed. Continuous cohomology can be computed using an algebra A(X) of differential forms. For fibrations it is shown that there is a Serre spectral sequence relating the continuous cohomology of base, fibre and total space.

For (graded commutative) algebras A satisfying an appropriate finiteness and nilpotency condition, a realization \(\Delta\) (A), which is a simplicial space, is defined. If A is free, the space of indecomposables of A is shown to be isomorphic to the space of continuous maps \(\pi_*(\Delta A)\to {\mathbb{R}}.\)

The real homotopy category is defined as the category of simplicial spaces (satisfying a nilpotency condition) localized with respect to maps inducing an isomorphism in continuous cohomology. The functor \(\Delta\) induces an equivalence between a homotopy category of algebras and the real homotopy category.

The paper contains as an interesting remark that the natural map \(\{\) rational homotopy types\(\}\) \(\to \{real\) homotopy types\(\}\) is neither injective nor surjective.

For (graded commutative) algebras A satisfying an appropriate finiteness and nilpotency condition, a realization \(\Delta\) (A), which is a simplicial space, is defined. If A is free, the space of indecomposables of A is shown to be isomorphic to the space of continuous maps \(\pi_*(\Delta A)\to {\mathbb{R}}.\)

The real homotopy category is defined as the category of simplicial spaces (satisfying a nilpotency condition) localized with respect to maps inducing an isomorphism in continuous cohomology. The functor \(\Delta\) induces an equivalence between a homotopy category of algebras and the real homotopy category.

The paper contains as an interesting remark that the natural map \(\{\) rational homotopy types\(\}\) \(\to \{real\) homotopy types\(\}\) is neither injective nor surjective.

Reviewer: M.Unsöld

### MSC:

55P60 | Localization and completion in homotopy theory |

55P62 | Rational homotopy theory |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

55P15 | Classification of homotopy type |

55U10 | Simplicial sets and complexes in algebraic topology |

55T10 | Serre spectral sequences |

### Keywords:

homotopy theory of simplicial spaces; topological groups; homotopy groups; fibration; continuous cohomology; differential forms; Serre spectral sequence; realization; space of indecomposables; real homotopy category; rational homotopy types; real homotopy types
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\textit{E. H. Brown jun.} and \textit{R. H. Szczarba}, Trans. Am. Math. Soc. 311, No. 1, 57--106 (1989; Zbl 0671.55006)

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