Rational Homotopy Type A Constructive Study via the Theory of the I*
This comprehensive monograph provides a self-contained treatment of the theory of I*-measure, or Sullivan's rational homotopy theory, from a constructive point of view. It centers on the notion of calculability which is due to the author himself, as are t
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1264 WuWen-tsun
Rational Homotopy Type A Constructive Study via the Theory of the I*-measure
Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Wu Wen-tsun Institute of Systems Science, Academia Sinica Beijing, 100080, P. R. China
Mathematics Subject Classification (1980): 57XX, 55XX, 58XX
ISBN 3-540-13611-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-13611-8 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, HemsbachfBergstr. 2146f3140-543210
PRE F A C
D. Sullivan has discovered the following remarkable fact: The differential graded algebra (abbreviated DGA) of differential forms on a differential manifold contains in formation not only on the infinite part of the cohomology ring of the manifold, but also on the infinite part of the homotopy ring under the Whitehead product. It was also D. Sullivan who extended the notion of DGA of differential forms to an arbitra ry simplicial complex. Moreover, he
introduced the notion of minimal model for such
a OGA, and showed that the same relations to homology and homotopy hold as in the ca se of manifolds; these relations can easily be deduced from such a minimal model. Though the main ideas of the theory may be traced back, as shown by Sullivan himself, to many predecessors from E. Cartan onwards, it is Sullivan who is the uncontestable founder of this whole theory which immensely influences further developments of algebraic topology. The present book aims at presenting the theory in an elementary form under the name of I"'measure which is a synonym of "rational homotopy type" or "minimal model" in current use. We adopt this name to give evidence of its measure theoretical Character,
with
particular
emphasis on its calculability. In fact Sul-
livan has expressed the opinion that one of the advantages of this theory lies in of fering "a new concreteness for calculation". We introduce here the notion of calcula bility of a certain measure with respect to a certain geometrical construction in a -k
technical SenSe. It is one of the aims of this book to show that the I measure is calculable with respect to most of the important geometrical constructions usually encountered in algebraic topology,
in contrast to the fact that even the simplest
classical cohomology H measure is not calculable, even with respect to the simplest geometrical constructions such as the additi
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