Cohomology rings of 3-local finite \(H\)-spaces (Q5957261)

Let \(X\) be a simply connected three-local finite \(H\)-space with \(H^*(X;\mathbb{F}_3)\) an associative ring. By the Borel structure theorem \(H^*(X; \mathbb{F}_3)\) is of the form NEWLINE\[NEWLINE\wedge(x_1,\dots, x_e) \otimes \mathbb{F}_3[y_1, \dots, y_k]/(y_1^{3f_1}, \dots,y_k^{3f_k})NEWLINE\]NEWLINE as an algebra where the degrees of \(x_i\) are odd and those of \(y_i\) are even. The purpose of this paper is to study the possible ring structures; e.g.NEWLINENEWLINENEWLINETheorem A: The even degree of generators of \(H^* (X; \mathbb{F}_3)\) lie between degree 8 and 20.NEWLINENEWLINENEWLINEThe Hopf algebra \(H^*(X; \mathbb{F}_3)\) is primitively generated. The proof of results about the algebra structure of \(H^* (X;\mathbb{F}_3)\) rely on information about the action of the Steenrod algebra on the odd degree generators.NEWLINENEWLINENEWLINETheorem E: If \(\overline x\in QH^{6n+3} (X; \mathbb{F}_3)\) and \({\mathcal P}^{3n+1} \overline x\in\text{im} {\mathcal P}^2\) then \({\mathcal P}^1\sigma^*(x) \in\text{im} {\mathcal P}^2\), where \(\sigma^*: QH^*(X;\mathbb{F}_3)\to PH^* (\Omega X;\mathbb{F}_3)\) is the cohomology suspension.