×

zbMATH — the first resource for mathematics

On the homotopy types of \(\mathrm{Sp}(n)\) gauge groups. (English) Zbl 1415.55003
For a compact connected simple Lie group \(G\) and an integer \(k\), let \(P_k\) be the principal \(G\)-bundle over \(S^4\) classified by the characteristic element \(k\in \mathbb{Z}= \pi_3(G)\), and let \(\mathcal{G}_k(G)\) denote the gauge group of \(P_k\). In this paper, the authors study the classification of homotopy types of gauge groups \(\{\mathcal{G}_k(G)\}_{k\in \mathbb{Z}}\) when \(G=\mathrm{Sp}(n)\). From now on, let us write \(\mathcal{G}_{k,n}=\mathcal{G}_k(\mathrm{Sp}(n))\), and let \((a,b)\) denote the greatest common divisor of the integers \(a\) and \(b\). Note that W. A. Sutherland [Proc. R. Soc. Edinb., Sect. A, Math. 121, No. 1–2, 185–190 (1992; Zbl 0761.55007)] previously showed that if \(\mathcal{G}_{k,n}\) and \(\mathcal{G}_{l,n}\) are homotopy equivalent, then \((k,n(2n+1))=(l,n(2n+1))\) for \(n\equiv 0 \mod 2\) and \((k,4n(2n+1))=(l,4n(n+1))\) for \(n\equiv 1 \mod 2\).
As the first step, the authors improve the above result and they prove that if \(\mathcal{G}_{k,n}\) and \(\mathcal{G}_{l,n}\) are homotopy equivalent, then \((k,4n(2n+1))=(l,4n(n+1))\).
Next they investigate the meaning of the number \(4n(2n+1)\). Let \(Q_n\) denote the quasiprojective space of rank \(n\) defined by I. M. James [Proc. Lond. Math. Soc. (3) 9, 115–140 (1959; Zbl 0089.39401)] and let \(\iota_n:Q_n\to\mathrm{Sp}(n)\) be the natural inclusion. Then they show that the order of the Samelson product \(\langle \epsilon ,\iota_n\rangle\) is \(4n(2n+1)\). Remark that the order of the Samelson product \(\langle \epsilon ,1_{\mathrm{Sp}(n)}\rangle\) is no less than that of \(\langle \epsilon ,\iota_n\rangle\) and they try to prove that both orders are equal. Although they cannot prove this, they show that it is equal if we localize them at the latter prime \(p\). More precisely, they show that \(\nu_p(|\langle \epsilon ,\iota_n\rangle |)=\nu_p(4n(2n+1))\) if \((p-1)^2+1\geq 2n\), where \(|x|\) is the order of the element \(x\).
As an application, they prove that \(\mathcal{G}_{k,n}\) and \(\mathcal{G}_{l,n}\) are \(p\)-local homotopy equivalent if and only if \(\nu_p((k,4n(2n+1))=\nu_p((l,4n(2n+1))\) for \((p-1)^2+1\geq 2n\). Moreover, by using a result due to E. M. Friedlander [Ann. Math. (2) 101, 510–520 (1975; Zbl 0308.55016)], they also show that \(\mathcal{G}_k(\mathrm{Spin} (2n+\epsilon))\) and \(\mathcal{G}_l(\mathrm{Spin} (2n+\epsilon))\) are \(p\)-local homotopy equivalent if and only if \(\nu_p(k,4n(2n+1))=\nu_p(l,4n(2n+1))\) when \((p-1)^2+1\geq 2n \geq 6\) and \(\epsilon =1,2\).

MSC:
55P15 Classification of homotopy type
54C35 Function spaces in general topology
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55Q15 Whitehead products and generalizations
PDF BibTeX Cite
Full Text: DOI
References:
[1] 10.1098/rsta.1983.0017 · Zbl 0509.14014
[2] 10.2307/2372495 · Zbl 0050.39603
[3] 10.1112/S0024611500012545 · Zbl 1024.55005
[4] 10.1016/j.topol.2018.01.005 · Zbl 1383.55005
[5] 10.2307/1970938 · Zbl 0308.55016
[6] 10.2307/1996373 · Zbl 0251.55018
[7] 10.1016/j.topol.2008.02.008 · Zbl 1144.55014
[8] 10.1017/S0308210500004480 · Zbl 1103.55004
[9] 10.2140/agt.2016.16.1813 · Zbl 1352.55005
[10] 10.1112/plms/s3-9.1.115 · Zbl 0089.39401
[11] 10.1017/S0017089507003606 · Zbl 1131.55008
[12] 10.3792/pjaa.86.15 · Zbl 1190.55007
[13] 10.1215/21562261-2693487 · Zbl 1321.55007
[14] 10.2140/agt.2013.13.1757 · Zbl 1276.57036
[15] 10.1017/S0308210512001278 · Zbl 1305.55005
[16] 10.1016/j.topol.2017.05.012 · Zbl 1373.55008
[17] 10.1112/jtopol/jtv025 · Zbl 1343.55003
[18] 10.1017/S0308210500024732 · Zbl 0722.55008
[19] 10.2140/pjm.1973.44.201 · Zbl 0248.55016
[20] 10.1215/kjm/1250280979 · Zbl 1170.55003
[21] 10.1017/S0308210500014220 · Zbl 0761.55007
[22] 10.1090/S0002-9947-07-04304-8 · Zbl 1117.55005
[23] 10.1215/0023608X-2010-005 · Zbl 1202.55004
[24] 10.2140/agt.2010.10.535 · Zbl 1196.55009
[25] ; Theriault, Osaka J. Math., 52, 15, (2015)
[26] 10.2140/agt.2017.17.1131 · Zbl 1361.55010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.