\(v_1\)-periodic 2-exponents of \(SU(2^e)\) and \(SU(2^e+1)\)
type: Item
instance of: scholarly article
title: \(v_1\)-periodic 2-exponents of \(SU(2^e)\) and \(SU(2^e+1)\)
zbMATH Open document ID: 1261.55007
author: Donald M. Davis
published in: Journal of Pure and Applied Algebra
publication date: 2012-10-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.4882
reviewed by: Behrooz Mashayekhy
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 55Q51
Mathematics Subject Classification ID: 11B73
zbMATH DE Number: 6094116
zbMATH Keywords: \(\nu_1\)-periodic homotopy groups
zbMATH Keywords: special unitary groups
zbMATH Keywords: homotopy 2-exponent
zbMATH Keywords: Stirling numbers
MaRDI profile type: Publication
OpenAlex ID: W1575347571
arXiv ID: 1109.4882
cites work: The ν 1 -Periodic Homotopy Groups of SU(n ) at Odd Primes
cites work: On the 2-primary \(v_{1}\)-periodic homotopy groups of spaces
cites work: The double suspension and exponents of the homotopy groups of spheres
cites work: The Hurewicz Map on Stunted Complex Projective Spaces
cites work: Divisibility by 2 and 3 of certain Stirling numbers
cites work: Divisibility by 2 of partial Stirling numbers
cites work: Q4010482
cites work: 2-primary v 1-periodic homotopy groups of SU(n) revisited
cites work: A number-theoretic approach to homotopy exponents of \(\text{SU}(n)\)
cites work: 2-primary exponents for the homotopy groups of spheres
DOI: 10.1016/J.JPAA.2012.01.001
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review text: Let \(n\) be a natural number and let \(\nu(n)\) denote the exponent of \(2\) in \(n\). Let NEWLINE\[NEWLINET_j(k)=\sum_{\mathrm{odd}\;i}{j\choose i}i^kNEWLINE\]NEWLINE denote a family of partial Stirling numbers. Suppose \(\mathbf{e}(k,n)=\min\{ \nu(T_j(k): j\geq n\}\) and \(\overline{\mathbf{e}}(n)= \max\{\mathbf{e}(k,n): k\in \mathbb{Z}\}\). The author mentions that \(\overline{\mathbf{e}}(n)\) is what we might call the \(\upsilon_1\)-periodic \(2\)-exponent of \(SU(n)\), where \(SU(n)\) is the special unitary group of degree \(n\).NEWLINENEWLINEThe main theorem of the paper under review which verifies a conjecture of the author's paper [Integers 8, No. 1, Article A56, 25 p., electronic only (2008; Zbl 1232.11034)] is as follows.NEWLINENEWLINETheorem. {\parindent=6mm\begin{itemize}\item[(a)] If \(e\geq 3\), then \(\mathbf{e}(k,2^e)\leq 2^e+2^{e-1}-1\) with equality occurring iff \(k\equiv 2^e-1\;\mathrm{mod}\;2^{2^{e-1}+e-1}\). \item[(b)] If \(e\geq 2\), then \(\mathbf{e}(k,2^e+1)\leq 2^e+2^{e-1}\) with equality occurring iff \(k\equiv 2^e+2^{2^{e-1}+e-1}\;\mathrm{mod}\;2^{2^{e-1}+e}\). NEWLINENEWLINE\end{itemize}} The homotopy \(2\)-exponent of a topological space \(X\), denoted \(\exp_2(X)\), is the largest \(k\) such that \(\pi_*(X)\) contains an element of order \(2^k\). An immediate consequence of the main theorem of the paper is as follows.NEWLINENEWLINECorollary. For \(\epsilon=\{0,1\}\) and \(2^e+\epsilon\geq 5\), NEWLINE\[NEWLINE \exp_2(SU(2^e+\epsilon))\geq 2^e+2^{e-1}-1+\epsilon .NEWLINE\]
description: scientific article; zbMATH DE number 6094116