I am reading now J. Stasheff's "Homotopy Associativity of H-Spaces, I":
https://www.jstor.org/stable/1993608.
I have following problem with Theorem 17, part of proof where it is shown that constructed space does not admit $A_p$-structure. It is written that
Thus we know that the Thom-Gysin sequence applies to the "Thom space", $C\mathcal{E}_i\cup_{\mathcal{p}_i}\mathcal{B}_i=XP(i)$. (...) In exactly the same way (as in real and complex projective spaces), since $H^*(X;\mathbb{Z}_p)=H^*(\mathbb{S}^{2n-1};\mathbb{Z}_p)$, we find that $H^*({XP(i);\mathbb{Z}_p})$ is a truncated polynomial algebra on a generator $u\in H^{2n}(XP(i);\mathbb{Z}_p)$ with $u^i\neq 0$.
I don't get it. In order to use the Thom-Gysin sequence or the Serre Spectral Sequence we have to know something about cohomology of total space, $\mathcal{E}_i$ in Stasheff's notation. We know that $\mathcal{E}_i$ is homotopy equivalent to $i$-fold join of $X$, and that $H^*(X;\mathbb{Z}_p)=H^*(\mathbb{S}^{2n-1};\mathbb{Z}_p)$. How can we show that $H^*(XP(i);\mathbb{Z}_p)$ is truncated polynomial algebra?