Equivalence classes of homotopy-associative comultiplications of finite complexes

Abstract

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, S n 1 + 1 V … V S n r + 1 . Two comultiplications of X are equivalent if there is a self-homotopy equivalence of X which carries one to the other. Let b ̃ a(X) , respectively b ̃ ac(X) , 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) n i ≠ n j + n k for every j, k with j < k and (b) n i ≠ 2 n j for every j with n j even, then b ̃ a(X) is finite. (2) b ̃ ac(X) is always finite. The methods of proof are algebraic and consist of a detailed examination of comultiplications of the free Lie algebra π #(ΩX) ⊗ 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.