On H-Spaces and a Congruence of Catalan Numbers

TL;DR

This paper explores the relationship between H-spaces, conjugacy classes in special unitary groups, and Catalan numbers, revealing new examples and structural insights in algebraic topology.

Contribution

It provides a novel enumeration of conjugacy classes related to Catalan numbers and characterizes when certain coset spaces admit p-local H-structures, including new examples.

Findings

Number of conjugacy classes is (1+C_{p-1})/p.

Only PSu(p) admits a p-local H-structure among certain coset spaces.

SU(4)/Z_3 is a new example of a globally defined H-space.

Abstract

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 $BSU(n)$.