# Groups of Homotopy Classes

## Rank formulas and homotopy-commutativity

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

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

Many of the sets that one encounters in homotopy classification problems have a natural group structure. Among these are the groups [A,nX] of homotopy classes of maps of a space A into a loop-space nx. Other examples are furnished by the groups ~(y) of homotopy classes of homotopy equivalences of a space Y with itself. The groups [A,nX] and ~(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,nX] and of C(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. Chapter 2 is purely algebraic. We recall the definition of the rank of a group and establish some of its properties. These facts, which may be found in the literature, are needed in later sections. Chapter 3 deals with the groups [A,nx] and the homomorphisms f*: [B,n~l ~ [A,nx] induced by maps f: A ~ B. We prove a general theorem on the rank of the intersection of coincidence subgroups (Theorem 3. 3).

Abelian group Finite Groups Homotopiegruppe Invariant Morphism algebra finite group homology homotopie homotopy theorem

- DOI https://doi.org/10.1007/978-3-662-15913-2
- Copyright Information Springer-Verlag Berlin Heidelberg 1964
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Springer Book Archive
- Print ISBN 978-3-662-15915-6
- Online ISBN 978-3-662-15913-2
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
- About this book