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Introduction

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Part of the Lecture Notes in Mathematics book series (LNM,volume 4)

Abstract

Many of the sets that one encounters in homotopy classification problems have a natural group structure. Among these are the groups [A,ΩX] of homotopy classes of maps of a space A into a loop-space ΩX. Other examples are furnished by the groups ξ(Y) of homotopy classes of homotopy equivalences of a space Y with itself. The groups [A,ΩX] and ξ(Y) are not necessarily abelian. It is our purpose to study these groups using a numerical invariant which can be defined for any group. This invariant, called the rank of a group, is a generalisation of the rank of a finitely generated abelian group. It tells whether or not the groups considered are finite and serves to distinguish two infinite groups. We express the rank of subgroups of [A,ΩX] and of ξ(Y) in terms of rational homology and homotopy invariants. The formulas which we obtain enable us to compute the rank in a large number of concrete cases. As the main application we establish several results on commutativity and homotopy-commutativity of H-spaces.

Keywords

  • Homotopy Class
  • Betti Number
  • Homotopy Type
  • Homotopy Group
  • Concrete Case

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1964 Springer-Verlag Berlin Heidelberg

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Arkowitz, M., Curjel, C.R. (1964). Introduction. In: Groups of Homotopy Classes. Lecture Notes in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-15913-2_1

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  • DOI: https://doi.org/10.1007/978-3-662-15913-2_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-15915-6

  • Online ISBN: 978-3-662-15913-2

  • eBook Packages: Springer Book Archive