Stable splittings of \(BP\) for some \(P\) of order thirty-two. (English) Zbl 0860.55016
Summary: We use the method of D. J. Benson and M. Feshbach obtained in [Topology 31, 157-176 (1992; Zbl 0752.55008)] to give the complete, 2-complete stable splitting of the classifying space \(BP\) for all but two of the groups \(P\) of order thirty-two which are not direct products in a nontrivial way. One of these two is \(\mathbb{Z}/32\), and \(B(\mathbb{Z}/32)\) is known to be indecomposable. For the other, \(P\) is the semidirect product of \((\mathbb{Z}/2)^4\) and \(\mathbb{Z}/2\). This is the only nonabelian group of order 32 which is not a direct product and has a subgroup isomorphic to \((\mathbb{Z}/2)^4\). We also give some results which refine the method of Benson and Feshbach. We have found that \(BP\) is indecomposable for three of these groups, and these are the first examples of indecomposable \(BP\) for which \(P\) is a nonabelian 2-group. Poincaré series are given for each classifying space using D. Rusin’s paper [Math. Comput. 53, No. 187, 359-385 (1989; Zbl 0677.20039)].
MSC:
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 55P42 | Stable homotopy theory, spectra |
References:
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