On the slice spectral sequence. (English) Zbl 1271.55015
This paper develops a variant of the slice spectral sequence of Hill, Hopkins and Ravenel. The slice spectral sequence is the spectral sequence of the slice filtration of the homotopy category of genuine \(G\)-spectra (where \(G\) is a finite group). The slice filtration is defined using the slice cells of dimension \(k\), which are the following collection of spectra: \(G_+ \wedge_H S^{n \rho_H}\) for \(n |H|=k\) and \(G_+ \wedge_H S^{n \rho_H-1}\) for \(n |H|-1=k\) (\(\rho_H\) is the real regular representation of \(H\), a subgroup of \(G\)).
In this paper the author considers the filtration obtained by using only those cells of the form: \(G_+ \wedge_H S^{n \rho_H}\) for \(n |H|=k\). The corresponding spectral sequence is called the regular slice spectral sequence. This variant is carefully analysed and compared to the original.
As an application two conjectures of Hill on slice towers are proven. The first concerns spectra whose homotopy groups are concentrated over a normal subgroup of \(G\). The second concerns the slices of Eilenberg-Mac Lane spectra. The paper concludes with some results relating the (co)connectivity a spectrum with the (co)connectivity of its slice tower.
In this paper the author considers the filtration obtained by using only those cells of the form: \(G_+ \wedge_H S^{n \rho_H}\) for \(n |H|=k\). The corresponding spectral sequence is called the regular slice spectral sequence. This variant is carefully analysed and compared to the original.
As an application two conjectures of Hill on slice towers are proven. The first concerns spectra whose homotopy groups are concentrated over a normal subgroup of \(G\). The second concerns the slices of Eilenberg-Mac Lane spectra. The paper concludes with some results relating the (co)connectivity a spectrum with the (co)connectivity of its slice tower.
Reviewer: David Barnes (Belfast)
MSC:
| 55T99 | Spectral sequences in algebraic topology |
| 55N91 | Equivariant homology and cohomology in algebraic topology |
| 55P91 | Equivariant homotopy theory in algebraic topology |
| 55Q91 | Equivariant homotopy groups |
References:
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| [3] | M A Hill, The Equivariant Slice Filtration: a Primer · Zbl 1403.55003 |
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