arXiv subjects
Publications and source records attributed to John R. Harper.
In 1960, J.F. Adams introduced secondary cohomology operations that are defined on cohomology elements on which sufficiently many Steenrod algebra elements vanish. This led to his theorem on singly generated cohomology rings, which in turn led to his celebrated resolution of the Hopf invariant one problem. Here we advertise a conjecture that would extend Adams' result and prove it in a special case.
For $p$ an odd prime and $F$ the cyclic group of order $p$, we show that the number of conjugacy classes of embeddings of $F$ in $SU(p)$ such that no element of $F$ has 1 as an eigenvalue is $(1+C_{p-1})/p$, where $C_{p-1}$ is a Catalan number. We prove that the only coset space $SU(p)/F$ that admits a $p$-local $H$-structure is the classical Lie group $PSU(p)$. We also show that $SU(4)/\mathbb Z_3$, where $\mathbb Z_3$ is embedded off the center of $SU(4)$, is a novel example of an $H$-space, even globally. We apply our results to the study of homotopy classes of maps from $BF$ to $BSU(n)$.