EQUIVALENCE CLASSES OF HOMOTOPY-ASSOCIATIVE COMULTIPLICATIONS OF FINITE COMPLEXES

Let X be a finite, 1-connected CW-complex which admits a homotopy-asso ciative comultiplication, Then X has the rational homology of a wedge of spheres, S-n1+1 V ... V S-nr+1. Two comultiplications of X are equi valent if there is a self-homotopy equivalence of X which carries one to the other, Let ($) over tilde b(a)(X), respectively ($) over tilde b(ac)(X), denote the set of equivalence classes of homotopy classes of homotopy-associative, respectively, homotopy-associative and homotopy -commutative, comultiplications of X. We prove the following basic fin iteness result: Theorem 6.1 (1) If for each i, (a) n(i) not equal n(j) + n(k) for every j, k with j < k and (b) n(i) not equal 2n(j) for eve ry j with n(j) even, then ($) over tilde b(a)(X) is finite. (2) ($) ov er tilde b(ac)(X) is always finite. The methods of proof are algebraic and consist of a detailed examination of comultiplications of the fre e Lie algebra pi(#)(Omega X) X Q. These algebraic methods and results appear to be of interest in their own right. For example, they provide dual versions of well-known results about Hopf algebras. In an append ix we show the group of self-homotopy equivalences that induce the ide ntity on all homology groups is finitely generated.