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Homotopy associativity of 𝐻-spaces. I

1963; American Mathematical Society; Volume: 108; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1963-99936-3

1088-6850

James Stasheff,

Fixed Point Theorems Analysis

JAMES DILLON STASHEFF(i) 1. Introduction.The concept of an //-space arose as a generalization of that of a topological group.The essential feature which is retained is a continuous multiplication with a unit.There is a significant class of spaces which are //-spaces but not topological groups.Some of the techniques which apply to topological groups can be applied to //-spaces, but not all.From the point of view of homotopy theory, it is not the existence of a continuous inverse which is the important distinguishing feature [6; 15], but rather the associativity of the multiplication.For example, if we regard S°, S1, S3 and S7 as the real, complex, quaternionic and Cayley numbers of unit norm, these spaces possess continuous multiplications, which in the first three cases are associative.Now it is possible to define real, complex and quaternionic projective spaces of arbitrarily large dimension, but this is not possible for the Cayley numbers.From the point of view of homotopy theory, we can investigate the "mechanism" which relates the associativity of the multiplication to the possible existence of projective spaces.First we consider the construction of the classical projective space as generalized by Milnor [8] for an arbitrary topological group (and further generalized by Dold and Lashof [3] for an arbitrary associative //-space).Given a topological group G, Milnor constructs fibre bundles p;:£;->B; with fibre G, the total space £¡ being the t'-fold join of G with itself.If G = Sd_1, d = 1,2,4, this gives the standard fibring of S"1-1 onto the corresponding projective space of dimension i -1.In the case of the Cayley numbers, only the fibrings of S7 onto a point and of S15 onto S8 can be constructed.It seems reasonable to ask whether something weaker than associativity might permit more but not all of these fibrings to be constructed.Sugawara [14] has shown that a variant of Milnor's construction can be carried one step further than for an arbitrary //-space if the multiplication is at least homotopy associative ; that is, if m : X x X -* X is the multiplication then the two maps of X x X x X into X given by the two ways of associating are homotopic, i.e., the diagram