Equivalence classes of homotopy-associative comultiplications of finite complexes Original Research Article

Let X be a finite, 1-connected CW-complex which admits a homotopy-associative comultiplication. Then X has the rational homology of a wedge of spheres, Sn1 + 1 V … V Snr + 1. Two comultiplications of X are equivalent if there is a self-homotopy equivalence of X which carries one to the other. Let image, respectively image, denote the set of equivalence classes of homotopy classes of homotopy-associative, respectively, homotopy-associative and homotopycommutative, comultiplications of X. We prove the following basic finiteness result: Theorem 6.1 (1) If for each i, (a) ni ≠ nj + nk for every j, k with j < k and (b) ni ≠ 2nj for every j with nj even, then image is finite. (2) image is always finite. The methods of proof are algebraic and consist of a detailed examination of comultiplications of the free Lie algebra π#(ΩX) circle times operator 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 appendix we show the group of self-homotopy equivalences that induce the identity on all homology groups is finitely generated.