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# Homology, Homotopy and Applications

## Volume 19 (2017)

### Number 2

### On $H$-spaces and a congruence of Catalan numbers

Pages: 21 – 30

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n2.a2

#### Authors

#### 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 $BSU(n)$.

#### Keywords

$H$-space, Catalan number, homotopy class of maps

#### 2010 Mathematics Subject Classification

05A15, 05E15, 11A07, 11B50, 55P45

Received 8 September 2016

Published 19 July 2017