On H-spaces and a Congruence of Catalan Numbers
Homology, Homotopy and Applications
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 + Cp−1)/p, where Cp−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)/Z3, where Z3 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).
Friedmann, Tamar and Harper, John, "On H-spaces and a Congruence of Catalan Numbers" (2017). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.