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Real homotopy theory of semi-algebraic sets. (English) Zbl 1254.14066

In this beautiful and difficult work the authors provide a detailed study of the theory of real homotopy type of semialgebraic sets. The present work completes the theory outlined some years ago by Kontsevich and Soibelman. The presentation of the theory is detailed and precise. After intuitive introduction, the authors review necessary facts from the semialgebraic geometry i.e. semialgebraic topology, Hauptvermutung and theory of currents. By the ingenious use of currents they define semialgebraic chains and complexes and by duality a commutative differential graded algebra of piecewise semialgebraic forms (CDGA for short). The main result of the paper is included in the Section 6, where the theorem on the existence of a zigzak natural transformations of the functor from the category of semialgebraic sets to the category of CDGA algebras and the famous Sullivan functor of piecewise polynomial forms, as defined in the work of A. K. Bousfield and V. K. A. M. Gugenheim [“On PL de Rham theory and rational homotopy type”, Mem. Am. Math. Soc. 179, 94 p. (1976; Zbl 0338.55008)]. The authors prove in fact, that for compact semialgebraic sets there exists a weak equivalence of categories. The theorem as it stands is sufficient for the applications, however the Authors suggest that it may be true without the compactness hypothesis.

MSC:

14P10 Semialgebraic sets and related spaces
55P62 Rational homotopy theory

Citations:

Zbl 0338.55008
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References:

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