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real homotopy theory

Contents

Idea

The authors of (Brown-Szczarba 95) show how the Bousfield-Gugenheim approach to rational homotopy theory may be extended to the framework of continuous cohomology. In the usual theory, the extension from the rationals to the reals has always been effected by a formal tensoring with the reals. While this allows a connection to de Rham theory, it has always been preferable to have a direct construction of real homotopy theory akin to the Postnikov tower-like construction of rational homotopy theory. Unfortunately, in the usual situation, such a construction is impossible due to the incompatibility of the algebra of \mathbb{R} (as an abelian group) with its natural topology. The authors of (Brown-Szczarba 95) remedy this problem by creating a theory of minimal models which incorporates the usual topology of the reals and a generalization of van Est's theorem? saying that H *(K(,n);)H^*(K(\mathbb{R},n); \mathbb{R}) is a freely generated algebra on one generator (a result which is not true in the discrete topology case). By using the natural topology of \mathbb{R}, there is an immediate connection to continuous cohomology and applications such as characteristic classes of foliations.

(from the Zentralblatt review of (Brown-Szczarba 95))

In (Brown-Szczarba 90) is established an equivalence between the localization of simplicial topological spaces at the real numbers and differential graded commutative algebras in the context of topological algebras.

References

  • Edgar Brown, Robert Szczarba, Real and Rational Homotopy Theory, in Handbook of Algebraic Topology. Elsevier, 1995 (Zentralblatt)

  • Edgar Brown, Robert Szczarba, Real and rational homotopy theory for spaces with arbitrary fundamental group Duke Mathematical Journal 71. (1993): 229-316.

  • Edgar Brown, Robert Szczarba, Continuous cohomology and Real homotopy type II Asterisque 191, Societe Mathematique De France (1990) (JSTOR)

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