Two-primary algebraic K-theory of pointed spaces

Rognes, John

doi:10.1016/s0040-9383(01)00005-2

Cited by 22 publications

(18 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The fiber of the cyclotomic trace map K(M U ) → T C(M U ) is equivalent to that of K(Z) → T C(Z), by [Du97], which now is quite well known [Ro02], [Ro03]. Our theorem therefore provides a key input to the computation of K(M U ).…”

Section: Introductionmentioning

confidence: 81%

Differentials in the homological homotopy fixed point spectral sequence

Bruner

Rognes

2005

Algebr. Geom. Topol.

Self Cite

View full text Add to dashboard Cite

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 E 2 s,t = H −s gp (T; H t (R; F p )), converging conditionally to the continuous homology H c s+t (R hT ; F p ) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations β ǫ Q i 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 E 2r -term of the spectral sequence there are 2r other classes in the E 2r -term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., 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 = M U , BP , ku, ko and tmf . Similar results apply for all finite subgroups C ⊂ T, 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.

show abstract

Section: Introductionmentioning

confidence: 81%

Differentials in the homological homotopy fixed point spectral sequence

Bruner

Rognes

2005

Algebr. Geom. Topol.

Self Cite

View full text Add to dashboard Cite

show abstract

“…As a sample application we show in 6.3 that for p ≥ 5 and M a compact smooth k -connected n-manifold with k ≥ 4p − 2 and n ≥ 12p − 5, the first p-torsion in the homotopy of the smooth concordance space C(M ) is π 4p−4 C(M) ( A 2-primary analog of this study was presented in [38]. Related results on the homotopy fiber of the linearization map L : A( * ) → K(Z) were given in [18].…”

Section: 2) W H( * ) σC ∨ (W H( * )/σC)mentioning

confidence: 88%

The smooth Whitehead spectrum of a point at odd regular primes

Rognes

2003

Geom. Topol.

Self Cite

View full text Add to dashboard Cite

Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K (Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum W h( * ) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S 1 -transfer map t : ΣCP ∞ → S . The homotopy groups of W h( * ) are determined in a range of degrees, and the cohomology of W h( * ) is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.

show abstract

“…Furthermore they obtained many strong nonvanishing results in [ABK72] about these groups of Gromoll. Their results perhaps combined with recent knowledge about a k (see [Rog02] and [Rog03]) should yield extremely substantial quantitative improvements to Theorem 3 and hence also to Theorem 1. We are very grateful to the referee for pointing out to us this direction for future investigation.…”

Section: Diffn //mentioning

confidence: 91%