Hahn, Jeremy; Wilson, Dylan Eilenberg-Mac Lane spectra as equivariant Thom spectra. (English) Zbl 1459.55008 Geom. Topol. 24, No. 6, 2709-2748 (2020). An important result by M. Mahowald [Topology 16, 249–256 (1977; Zbl 0357.55020)] states that the Eilenberg-Mac Lane spectrum \(H\mathbb F_2\) is the Thom spectrum of a certain double loop map \(\Omega^2 S^3 \to BO\). In the paper under review, the authors establish various equivariant refinements of this result. One main theorem states that for \(G = C_{2^n}\) the cyclic group of order \(2^n\) and for \(\lambda\) the standard representation of \(G\) on \(\mathbb C\), there is a \(G\)-action on the quaternionic projective space \(\mathbb HP^{\infty}\) and an \(\Omega^{\lambda+1}\)-map \(\Omega^{\lambda+1} \mathbb HP^{\infty} \to BO_G\) whose Thom spectrum is \(H\underline{\mathbb F}{}_2\), the Eilenberg-Mac Lane spectrum on the constant \(G\)-Mackey functor \(\underline{\mathbb F}{}_2\). This result does not only refine Mahowald’s theorem, but also the observation (which the authors attribute to Hopkins) that \(\Omega^2 S^3 \to BO\) admits a triple delooping. The \(G = C_2\)-case of this main theorem generalizes a result by M. Behrens and D. Wilson [Proc. Am. Math. Soc. 146, No. 11, 5003–5012 (2018; Zbl 1409.55010)]. Hopkins also gave a description of the Eilenberg-Mac Lane spectrum \(H\mathbb F_p\) for an odd prime \(p\) as the Thom spectrum of a map \(\Omega^{2}S^3 \to B\mathrm{GL}_1(S^0_{(p)})\) to the classifying space of the units of the \(p\)-local sphere spectrum. As another main theorem of the paper under review, the authors give an equivariant refinement of this result by exhibiting \(H\underline{\mathbb F}{}_p\) as the Thom spectrum of a suitable map \(\Omega^{\lambda} S^{\lambda +1} \to B\mathrm{GL}_1(S^0_{(p)})\). There is a \(p\)-local integral variant of the latter result. The authors also explain why the additional delooping in the case \(p=2\) cannot exist in the odd primary case: they show that in contrast to the situation at the prime \(2\), for an odd prime \(p\) there is no triple loop map to \(B\mathrm{GL}_1(S^0_{(p)})\) with Thom spectrum \(H\mathbb F_p\). Reviewer: Steffen Sagave (Nijmegen) Cited in 2 ReviewsCited in 3 Documents MSC: 55P91 Equivariant homotopy theory in algebraic topology 55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) Keywords:equivariant Thom spectrum; Eilenberg-Mac Lane spectrum Citations:Zbl 0357.55020; Zbl 1409.55010 PDFBibTeX XMLCite \textit{J. Hahn} and \textit{D. Wilson}, Geom. Topol. 24, No. 6, 2709--2748 (2020; Zbl 1459.55008) Full Text: DOI arXiv References: [1] 10.1112/topo.12084 · Zbl 1417.55007 [2] 10.4310/jdg/1214428815 · Zbl 0215.24403 [3] 10.1016/j.aim.2016.08.043 · Zbl 1348.18020 [4] 10.1090/proc/14175 · Zbl 1409.55010 [5] 10.2140/gt.2010.14.1165 · Zbl 1219.19006 [6] 10.1016/j.aim.2015.07.013 · Zbl 1329.55012 [7] 10.1016/0040-9383(73)90014-1 · Zbl 0266.55012 [8] 10.2307/1997768 · Zbl 0404.55003 [9] 10.1215/ijm/1256047369 [10] 10.1007/BF01391176 · Zbl 0439.55002 [11] 10.2307/2001770 · Zbl 0769.54041 [12] 10.1007/BF01229795 · Zbl 0334.55004 [13] 10.1007/BFb0085965 [14] 10.1007/BFb0083002 [15] 10.1007/BFb0099250 [16] 10.4007/annals.2016.184.1.1 · Zbl 1366.55007 [17] 10.1017/S0305004199003588 · Zbl 0932.55015 [18] 10.2307/1969666 · Zbl 0077.36502 [19] 10.24033/asens.1163 · Zbl 0194.24101 [20] 10.1007/BFb0059024 [21] 10.1007/978-3-540-79890-3 · Zbl 1153.55005 [22] 10.2140/agt.2018.18.2541 · Zbl 1402.55003 [23] 10.4310/HHA.2010.v12.n1.a7 · Zbl 1195.55014 [24] 10.1007/BFb0075778 · Zbl 0611.55001 [25] 10.1515/9781400830558 · Zbl 1175.18001 [26] 10.1016/0040-9383(77)90005-2 · Zbl 0357.55020 [27] 10.1215/S0012-7094-79-04628-3 · Zbl 0418.55012 [28] 10.1007/978-3-0348-8312-2_16 [29] 10.1112/jtopol/jtv021 · Zbl 1335.55009 [30] 10.2140/gt.2016.20.3133 · Zbl 1373.14008 [31] 10.1007/BF01214823 · Zbl 0521.55011 [32] 10.2307/2042089 · Zbl 0385.55009 [33] 10.1090/S0894-0347-06-00521-2 · Zbl 1106.55002 [34] 10.1112/S0024610700001241 · Zbl 1024.55009 [35] 10.1017/9781108349161 · Zbl 1451.55001 [36] 10.1007/BF02684593 · Zbl 0199.26202 [37] ; Segal, Actes du Congrès International des Mathématiciens, II, 59 (1971) [38] 10.1007/BF00535647 · Zbl 0790.55006 [39] 10.1017/S0305004100035878 [40] 10.2307/1998063 · Zbl 0447.55011 [41] 10.2307/1998768 · Zbl 0546.55023 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.