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The homology of iterated loop spaces

The homology of iterated loop spaces Abstract. The chain complex of the iterated loop space is expressed through the chain complex over an initial space in terms of the cobar construction. The corresponding spectral sequence is constructed and its ®rst term is calculated. Applications to the calculation of the homology of the iterated loop spaces of the stunted real and complex projective spaces are given. The origin of this work is in some machine computations undertaken by Sergeraert; a brief report about them is given in the appendix. 1991 Mathematics Subject Classi®cation: 55P35, 55P48, 55-04. Introduction In recent years, to solve various problems in Algebraic Topology it has been necessary to consider more and more complicated structures on the singular chain complex CÃ X of a topological space X and its homology HÃ X . One of the most di½cult problem is the problem of calculating the homology groups of iterated loop spaces. The ®rst steps toward solving this problem were made by J. F. Adams, [1]. To calculate the homology HÃ X of the loop space X of a topological space X he introduced the notion of the cobar construction FK on a coalgebra K. Recall that a chain complex K is called http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

The homology of iterated loop spaces

Forum Mathematicum , Volume 14 (3) – Apr 15, 2002

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References (24)

Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2002.016
Publisher site
See Article on Publisher Site

Abstract

Abstract. The chain complex of the iterated loop space is expressed through the chain complex over an initial space in terms of the cobar construction. The corresponding spectral sequence is constructed and its ®rst term is calculated. Applications to the calculation of the homology of the iterated loop spaces of the stunted real and complex projective spaces are given. The origin of this work is in some machine computations undertaken by Sergeraert; a brief report about them is given in the appendix. 1991 Mathematics Subject Classi®cation: 55P35, 55P48, 55-04. Introduction In recent years, to solve various problems in Algebraic Topology it has been necessary to consider more and more complicated structures on the singular chain complex CÃ X of a topological space X and its homology HÃ X . One of the most di½cult problem is the problem of calculating the homology groups of iterated loop spaces. The ®rst steps toward solving this problem were made by J. F. Adams, [1]. To calculate the homology HÃ X of the loop space X of a topological space X he introduced the notion of the cobar construction FK on a coalgebra K. Recall that a chain complex K is called

Journal

Forum Mathematicumde Gruyter

Published: Apr 15, 2002

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