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The slices of \(S^n \wedge H \underline{\mathbb{Z}}\) for cyclic \(p\)-groups. (English) Zbl 1394.55007
The slice filtration is a filtration of equivariant spectra which was developed by M. A. Hill et al. [Ann. Math. (2) 184, No. 1, 1–262 (2016; Zbl 1366.55007)] in their solution to the Kervaire invariant one problem as one of the landmark papers in algebraic topology. Even though the slice tower of a \(G\)-spectrum is an equivariant analogue of the Postnikov tower of a path-connected space, unlike the fibers from the Postnikov tower, slices need not be Eilenberg-MacLane spectra. The slice tower for certain Eilenberg-MacLane spectra was given by M. A. Hill [Homology Homotopy Appl. 14, No. 2, 143–166 (2012; Zbl 1403.55003)] and for more general Eilenberg-MacLane spectra by J. Ullman [Algebr. Geom. Topol. 13, No. 3, 1743–1755 (2013; Zbl 1271.55015)].
In this paper, the author explicitly describes the slice tower for all \(G\)-spectra of the form \(S^n \wedge H\underline{\mathbb{Z}}\), where \(n\) is a nonnegative integer, and \(G\) is a cyclic \(p\)-group \(C_{p^k}\) for \(p\) an odd prime. More precisely, she shows that the slice sections \(P^m (S^n \wedge H\underline{\mathbb{Z}})\) are of the form \(S^W \wedge H\underline{\mathbb{Z}}\), where \(W\) is a \(C_{p^k}\)-representation of dimension \(n\) with \(n \leq m \leq (n-2)p^k -1\) and \(S^W\) is a representation sphere. The author also proves that the nontrivial slices \(P_m^m (S^n \wedge H\underline{\mathbb{Z}})\) are of the form \(S^V \wedge H\underline{B}_{i,j}\), where \(\underline{B}_{i,j}\) is a \(C_{p^k}\)-Mackey functor and \(V\) is a \(C_{p^k}\)-representation of dimension \(m\) with \(m \equiv -1 ~(\text{mod}~ p)\) and \(n \leq m \leq (n-2)p^k -1\).

MSC:
55N91 Equivariant homology and cohomology in algebraic topology
55P91 Equivariant homotopy theory in algebraic topology
55P42 Stable homotopy theory, spectra
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