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2005 Differentials in the homological homotopy fixed point spectral sequence
Robert R Bruner, John Rognes
Algebr. Geom. Topol. 5(2): 653-690 (2005). DOI: 10.2140/agt.2005.5.653


We analyze in homological terms the homotopy fixed point spectrum of a T–equivariant commutative S–algebra R. There is a homological homotopy fixed point spectral sequence with Es,t2=Hgps(T;Ht(R;Fp)), converging conditionally to the continuous homology Hs+tc(RhT;Fp) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations βϵQi acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E2r–term of the spectral sequence there are 2r other classes in the E2r–term (obtained mostly by Dyer–Lashof operations on x) that are infinite cycles, ie survive to the E–term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R=THH(B) of many S–algebras, including B=MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups CT, and for the Tate and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K–theory of commutative S–algebras.


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Robert R Bruner. John Rognes. "Differentials in the homological homotopy fixed point spectral sequence." Algebr. Geom. Topol. 5 (2) 653 - 690, 2005.


Received: 2 June 2004; Revised: 3 June 2005; Accepted: 21 June 2005; Published: 2005
First available in Project Euclid: 20 December 2017

zbMATH: 1078.19003
MathSciNet: MR2153113
Digital Object Identifier: 10.2140/agt.2005.5.653

Primary: 19D55, 55S12, 55T05
Secondary: 55P43, 55P91

Rights: Copyright © 2005 Mathematical Sciences Publishers


Vol.5 • No. 2 • 2005
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