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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
##### Keywords:
slice filtration; equivariant homotopy
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