On \(H\)-spaces and a congruence of Catalan numbers

DOI10.4310/HHA.2017.V19.N2.A2zbMATH Open1377.05009arXiv1612.03837MaRDI QIDQ1689726

Peng Zhang

Publication date: 17 January 2018

Published in: Homology, Homotopy and Applications (Search for Journal in Brave)

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 $BSU(n)$.

Full work available at URL: https://arxiv.org/abs/1612.03837

zbMATH Keywords

Catalan number\(H\)-spacehomotopy class of maps

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) (H)-spaces and duals (55P45) Congruences; primitive roots; residue systems (11A07) Sequences (mod (m)) (11B50)

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