The homology algebra of finite H-spaces
Let \(\mu\) : \(X\times X\to X\) be the multiplication of an H-space X, H \(*=H\) *(X,\({\mathbb{Z}}/(p))\), \(H_*=H_*(X,{\mathbb{Z}}/(p))\). The prime p is assumed to be odd, X simply connected, \(H_*\) finite with associative Pontrjagin product. Finally, it is assumed that x \(p=0\) for all x in the augmentation ideal \(\tilde H\) *. The paper studies the implications of the action of the Steenrod algebra \(A=A\) *(p) of all stable cohomology operations with \({\mathbb{Z}}/(p)\) coefficients on the module of indecomposables QH * on the Lie algebra structure of \(PH_*\), the primitive elements in \(H_*\). Here is a sample result (Theorem 1.2): Suppose \(p=3\). a) If rank X\(<8\), then A acts trivially on \(Q^{even}H\) *. b) If rank X\(=8\) and A acts non-trivially on \(Q^{even}H\) *, H *\(\approx H\) \(*(E_ 8,{\mathbb{Z}}/(3))\) as Hopf algebras over A.
primitive elements of the homology coalgebra
scientific article; zbMATH DE number 4034689
scientific article