A nonexterior Hopf algebra and loop spaces (Q1924645)

Let \(A\) be the Hopf algebra over \(\mathbb{Z}/p\) given by \(A= \Lambda (x_i, y_i \mid 0 \leq i \leq p-2) \otimes (\mathbb{Z}/p [z]/z^p)\), with \(\deg x_i = 2(p+1) i+3\), \(\deg y_i= 2(p+1)\) \((i +1) -1\) and \(\deg z= 2(p+1)\). If \(p= 2,3,5\), then \(A= H^* (X; \mathbb{Z}/p)\) with \(X=G_2\), \(F_4\), \(E_8\). In this paper, the author shows that there is no simply connected loop space \(X\) with the homotopy type of a CW-complex of finite type so that \(H^*(X; \mathbb{Z}/p) \cong A\), when \(p \geq 7\).