Groups of Homotopy Classes pp 1-2 | Cite as

# Introduction

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## 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.