Equivalence classes of homotopy-associative comultiplications of finite complexes
For \(X\) a topological space the authors consider various sets of comultiplications on \(X\). They denote by \(\mathbb{C}(X)\) the set of homotopy classes of comultiplications on \(X\), by \(\mathbb{C}_a(X)\) respectively \(\mathbb{C}_{ac} (X)\) the set of homotopy classes of homotopy-associative comultiplications, respectively of both homotopy-associative and homotopy-commutative comultiplications of \(X\) and by \(\widetilde\mathbb{C} (X)\), \(\widetilde \mathbb{C}_a (X)\), \(\widetilde\mathbb{C}_{ac} (X)\) the corresponding equivalence classes. This paper has six sections and an appendix. After establishing basic facts concerning comultiplications and rationalization they obtain information about comultiplications of a space \(X\) by studying comultiplications of the rational homotopy Lie algebra \(\pi_\# (\Omega X) \otimes Q\) of \(X\). In the third section the authors relate these notions and conclude that \(\widetilde\mathbb{C} (X)\), \(\widetilde\mathbb{C}_a (X)\), \(\widetilde C_{ac} (X)\) are finite whenever the corresponding equivalence classes of Lie algebra comultiplications of \(\pi_\# (\Omega X) \otimes Q\) are finite. There are also results in this paper interesting in their own right as the theorem which asserts that an associative comultiplication on a Lie algebra is determined by its quadratic part. The main result in this paper concerns a finite complex \(X\) which admits a homotopy-associative comultiplication. A finite co-\(H\)-space \(X\) has the rational homotopy type of a wedge of spheres \(X_Q \equiv S_Q^{n_1+1} \vee\dots \vee S_Q^{n_r+1}\) for integers \(n_1\leq \cdots \leq n_r\) (so \((n_1, \dots, n_r)\) is the type of \(X)\). They prove that if for each \(i\) (a) \(n_i \neq n_j + n_k\) for every \(j,k\) with \(j<k\) and (b) \(n_i\neq 2n_j\) for every \(j\) with \(n_j\) even then \(\widetilde\mathbb{C}_a (X)\) is finite. Also they prove that \(\widetilde\mathbb{C}_{ac} (X)\) is finite. In the last section the authors give several examples that illustrate their results. In the appendix they prove for a finite complex that the group of self-homotopy equivalences that induce the identity on homology groups is finitely generated. All spaces in this paper are 1-connected based space of the homotopy type of a based CW complex or the rationalization of such space.
scientific article; zbMATH DE number 810229
scientific article