On Samelson products in $SU(n)$-and $Sp(n)$

TL;DR

The paper advances the understanding of Samelson products in classical Lie groups by giving explicit odd-primary orders for many cases of the products in SU($n$) and Sp($n$). It develops an unstable $K$-theory framework and uses fibration and cohomology techniques to compute these orders for $a+b=n$ and for $a+b=n+k$ with small $k$, yielding precise torsion factors. Two key applications are provided: a new proof of homotopy commutativity of $SU(n)_{(p)}$ for $n=3,4,5$ at primes $p>2n$, and a concrete upper bound on the number of $p$-local gauge-types for principal $SU(4)$-bundles, via the order of the odd-primary commutator. The paper also analyzes the group $[\Sigma^{4m-1}Q_{n-m+1}, Sp(n)]$, giving an exact order formula depending on the parity of $m$, and tying these results to the symplectic $K$-theory and quasi-projective spaces.

Abstract

Let $a$ and $b$ be two positive integers such that $a, b < n$. We denote the inclusion $Σ\mathbb{C}P^a\rightarrow SU(n)$ by $\varepsilon_{a,n}$. Also, let $m$ and $n$ be two positive integers such that $m < n$. This article has two parts. In the first part, we will study the order of the Samelson product $\langle \varepsilon_{a,n}, \varepsilon_{b,n}\rangle$ where $a+b=n+k$, for $k \geq 0$. Also, we will give two applications. In the second part, we will study the order of the Samelson product $S^{4m-1}\wedge Q_{n-m+1}\rightarrow Sp(n)$, where $Q_{n-m+1}$ is the symplectic quasi-projective space of rank $n-m+1$.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof