The p-completed cyclotomic trace in degree 2

(English) The goal of this thesis has been to represent the p-adic valuation of the special values of p-adic L-functions via Euler Characteristics using different cohomology theories in various settings. More specifically, homotopy theoretic invariants such as the K(1)-local K-theory spectrum have been used to describe these values in the setting of commutative and noncommutative totally real fields and also in some cases for imaginary quadratic fields. The most successful case has been in the commutative and totally real case, where the padic valuation of the Deligne-Ribet p-adic L-function can now be understood using étale cohomology, K(1)-local K-theory, and also as the fiber of the (K(1)-local) cyclotomic trace map, therefore extending the work of Hesselholt, which only covered F = Q. In addition to this, the foundation has been layed to extend these results in other directions, such as in the case of function fields, or more precisely, to the case of Abelian varieties with semistable reduction over function fields. (Català) L'objectiu d'aquesta tesi ha estat representar la valoració p-àdica dels valors especials de funcions-L p-àdiques mitjançant característiques d'Euler utilitzant diferents teories de cohomologia en diversos contextos. Més específicament, s'han utilitzat invariants homotòpics com l'espectre K(1)-local de K-teoria per descriure aquests valors en el context de cossos reals totalment commutatius i no-commutatius, i també en alguns casos per a cossos quadràtics imaginaris. El cas més exitós ha estat el commutatiu i totalment real, on ara es pot entendre la valoració p-àdica de la funció-L p-àdica de Deligne-Ribet utilitzant cohomologia ètale, K-teoria K(1)-local, i la fibra del mapa de traça ciclotòmica (K(1)-local), estenent així el treball de Hesselholt, que només cobreix el cas F = Q. Finalment, s'ha establert la base per estendre aquests resultats en altres direccions, com ara en el cas de cossos de funcions, o més concretament, en el cas de varietats abelianes amb reducció semiestable sobre cossos de funcions.

If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace from K(A) to TC(A) is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision. The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and - more relevantly for our current application - the T-Tate spectrum of topological Hochschild homology, where T is the circle group

Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $\infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $\varphi_p: X\to X^{tC_p}$ for all primes $p$. Here $X^{tC_p}=\mathrm{cofib}(\mathrm{Nm}: X_{hC_p}\to X^{hC_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology. In order to construct the maps $\varphi_p: X\to X^{tC_p}$ in the example of topological Hochschild homology we introduce and study Tate diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular we prove a version of the Segal conjecture for the Tate diagonals and relate these Frobenius homomorphisms to power operations.

The topological Hochschild homology THH(R) of a commutative S-algebra (E_infty ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Boekstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K-theory of S-algebras, by means of the cyclotomic trace map to topological cyclic homology.

We describe a construction of the cyclotomic structure on topological Hochschild homology ( THH ) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology ( TC ) and TR -theory: one starting with a ring C_n -spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum A . We describe spectral sequences computing the relative theory over A in terms of TR over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on TR and TC .

The talks covered advances in algebraic K -theory and topological cyclic homology, geometric group theory, low dimensional topology relying on a mixture of combinatorial and analytic methods, classification of high-dimensional manifolds and more. Special emphasis was given to a recent breakthrough on the question of triangulability of high-dimensional manifolds.

Topological cyclic homology of the integers at two

Let L be a finite extension of Q p with ring of integers O L .We show that periodic topological cyclic homology of O L , over the base E -ring S W (Fq) [z] carries a p-height one formal group law mod (p) that depends on an Eisenstein polynomial of L over Q p for a choice of uniformizer O L .

Publisher Correction: Topological cyclic homology of local fields

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 beta^epsilon 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 $^{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^infty-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 C of 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.