Topology of unitary groups and the prime orders of binomial coefficients

TL;DR

This paper investigates the topology of unitary groups by analyzing the induced homomorphism on cohomology from the quotient map of SU(n) to PSU(n), using prime orders of binomial coefficients.

Contribution

It provides a detailed computation of the induced cohomology homomorphism for the quotient map, revealing the role of prime orders of binomial coefficients in the topology of unitary groups.

Findings

Explicit description of the induced homomorphism on cohomology.

Identification of prime order factors of binomial coefficients affecting topology.

Enhanced understanding of the cohomological structure of PSU(n).

Abstract

Let $c:SU(n)\rightarrow PSU(n)=SU(n)/\mathbb{Z}_{n}$ be the quotient map of the special unitary group $SU(n)$ by its center subgroup $\mathbb{Z}_{n}$. We determine the induced homomorphism $c^{\ast}:$ $H^{\ast}(PSU(n))\rightarrow H^{\ast}(SU(n))$ on cohomologies by computing with the prime orders of binomial coefficients