Techniques, computations, and conjectures for semi-topological K-theory

Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectral sequence of an //-space is one example that has been investigated by many authors [3; 1; 7; 8]. Borel [3] and Araki [1] proved algebraic theorems about the structure of differential Hopf algebras of special kinds. These special theorems enabled them to determine the odd torsion in the cohomology of the exceptional Lie groups. If X and Tare //-spaces and/:X-> Tis a fibre map which is multiplicative, then the spectral sequence of / is a spectral sequence of Hopf algebras. This situation was first discussed by J. C. Moore [17], and later by the author [5; 6]. The techniques of [5] were later extended by the author (in unpublished work) to prove theorems about the homology and cohomology suspensions, i.e., when X is the space of paths of the //-space Y. The proofs rested upon a general theorem about the structure of this spectral sequence. Some of these suspension theorems had been proved by Moore using a different spectral sequence of Hopf algebras [12]. In this paper we make a study of differential Hopf algebras, and prove general theorems on the structure of their homology. These theorems generalize the results of Borel and Araki. Applied to the case of multiplicative fibre maps, we obtain a general theorem about the structure of the spectral sequence (even in the nonacyclic case), which, in particular, yields simple proofs of the suspension theorems mentioned above. Applied to the Bockstein spectral sequence, we get information on torsion in //-spaces. This study of differential Hopf algebras depends on two spectral sequences which may be defined in different circumstances. If one of them is defined, then the terms of that spectral sequence satisfy the conditions necessary for the other to be defined. Thus we get a spectral sequence for the term of the other spectral sequence(2), and this spectral sequence has a very simple form which makes it easy to calculate the form of its homology. Thus the structure of the limit

It is shown that the Grayson tower for$K$-theory of smooth algebraic varieties is isomorphic to the slice tower of$S^{1}$-spectra. We also extend the Grayson tower to bispectra, and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic$K$-theory spectrum$\mathit{KGL}$. This solves Suslin’s problem about these two spectral sequences in the affirmative.

In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its E 2 E_2 -page. Our description of the E 2 E_2 -page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost’s cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work.

For a pointed cosimplicial spaceX •, the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX •. In this paper,X • is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here.

the technique of obstruction theory developed in the last chapter focuses attention on the computation of homotopy groups. In this chapter we obtain some results about the homotopy groups of spheres. The method we follow is due to Serre1 and uses the technical tool known as a spectral sequence. This algebraic concept is introduced for the study of the homology and cohomology properties of arbitrary fibrations, but it has other important applications in algebraic topology, and the number of these is constantly increasing. Some indication of the power of spectral sequences will be apparent from the results obtained by its use here.

Moss’ theorem, which relates Massey products in the $$E_r$$ E r -page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric monoidal stable simplicial model category, we prove a general version of Moss’ theorem which applies to spectral sequences that arise from filtrations compatible with the monoidal structure. This involves the study of Massey products and Toda brackets in a non-strictly associative context. The theorem has broad applications, e.g., to the computation of the motivic slice spectral sequence and other colocalization towers.

In this paper, we introduce a four-filtrated version of the May spectral sequence (MSS), from which we study the general properties of the spectral sequence and give a collapse theorem. We also give an efficient method to detect generators of May E1-term E1s,t,b,* for a given (s, t, b, *). As an application, we give a method to prove the non-triviality of some compositions of the known homotopy elements in the classical Adams spectral sequence (ASS).

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.

Spectral sequences in combinatorial geometry: Cheeses, inscribed sets, and Borsuk–Ulam type theorems

The nerve of a family of sets is a simplicial complex that records the intersection pattern of its subfamilies. Nerves are widely used in computational geometry and topology, because the nerve theorem guarantees that the nerve of a family of geometric objects has the same topology as the union of the objects, if they form a good cover. In this paper, we relax the good cover assumption to the case where each subfamily intersects in a disjoint union of possibly several homology cells, and we prove a generalization of the nerve theorem in this framework, using spectral sequences from algebraic topology. We then deduce a new topological Helly-type theorem that unifies previous results of Amenta, Kalai and Meshulam, and Matousek. This Helly-type theorem is used to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory.

In this paper, we establish explicit formulas for the differentials in the homology Atiyah–Hirzebruch spectral sequences computing homotopy groups and [Formula: see text] ([Formula: see text])-homology groups. While the differentials in the [Formula: see text]-cohomology Atiyah–Hirzebruch spectral sequence are well-documented, their direct adaptation to the [Formula: see text]-homology spectral sequence remains non-obvious. In this work, we prove that the differentials in the [Formula: see text]-homology spectral sequence admit analogous expressions to their [Formula: see text]-cohomology counterparts, thereby completing this correspondence.

We describe explicitly the Voevodsky's triangulated category of motives $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$ (and give a ‘differential graded enhancement’ of it). This enables us to able to verify that DMgm ℚ is (anti)isomorphic to Hanamura's $\mathcal{D}$(k).We obtain a description of all subcategories (including those of Tate motives) and of all localizations of $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$. We construct a conservative weight complex functor $t:\smash{\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}}\to\smash{K^{\mathrm{b}}(\operatorname{Chow}^{\mathrm{eff}})}$; t gives an isomorphism $K_0(\smash{\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}})\to\smash{K_0(\operatorname{Chow}^{\mathrm{eff}})}$. A motif is mixed Tate whenever its weight complex is. Over finite fields the Beilinson–Parshin conjecture holds if and only if tℚ is an equivalence.For a realization D of $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$ we construct a spectral sequence S (the spectral sequence of motivic descent) converging to the cohomology of an arbitrary motif X. S is ‘motivically functorial’; it gives a canonical functorial weight filtration on the cohomology of D(X). For the ‘standard’ realizations this filtration coincides with the usual one (up to a shift of indices). For the motivic cohomology this weight filtration is non-trivial and appears to be quite new.We define the (rational) length of a motif M; modulo certain ‘standard’ conjectures this length coincides with the maximal length of the weight filtration of the singular cohomology of M.

Algebraic K -theory and the related motivic cohomology are a systematic way of producing invariants for algebraic or geometric structures. Its definition and methods are taken from algebraic topology, but it has also proved particularly fruitful for problems of algebraic geometry, number theory or quadratic forms. 19 one-hour talks presented a wide range of results on K -theory itself and applications. We had a lively evening session trading questions and discussing open problems.

The Adams spectral sequence has been an important tool in research on the stable homotopy of the spheres.In this note we outline new information about a variant of the Adams sequence which was introduced by Novikov [7].We develop simplified techniques of computation which allow us to discover vanishing lines and periodicity near the edge of the E 2 -term, interesting elements in E^'*, and a counterexample to one of Novikov's conjectures.In this way we obtain independently the values of many low-dimensional stems up to group extension.The new methods stem from a deeper understanding of the Brown-Peterson cohomology theory, due largely to Quillen [8]; see also [4].Details will appear elsewhere; or see [ll].When p is odd, the p-primary part of the Novikov sequence behaves nicely in comparison with the ordinary Adams sequence.Computing the £ 2 -term seems to be as easy, and the Novikov sequence has many fewer nonzero differentials (in stems ^45, at least, if p = 3), and periodicity near the edge.The case p = 2 is sharply different.Computing E 2 is more difficult.There are also hordes of nonzero differentials dz, but they form a regular pattern, and no nonzero differentials outside the pattern have been found.Thus the diagram of £4 ( =£oo in dimensions ^17) suggests a vanishing line for E w much lower than that of £2 of the classical Adams spectral sequence [3].It is a pleasure to thank Arunas Liulevicius, my thesis adviser, for his help.In particular, parts of the proofs of Proposition 1 and Theorem 7 are due to him.I am also grateful to many others for their suggestions, and especially to Frank Adams.1.The spectral sequence.The construction of the classical Adams spectral sequence for the spheres [l ] works equally well if the spec-

Let [Formula: see text] be a [Formula: see text]-equivariant algebraic principal [Formula: see text]-bundle over a normal complex affine variety [Formula: see text] equipped with an action of [Formula: see text], where [Formula: see text] and [Formula: see text] are complex linear algebraic groups. Suppose [Formula: see text] is contractible as a topological [Formula: see text]-space with a dense orbit, and [Formula: see text] is a [Formula: see text]-fixed point. We show that if [Formula: see text] is reductive, then [Formula: see text] admits a [Formula: see text]-equivariant isomorphism with the product principal [Formula: see text]-bundle [Formula: see text], where [Formula: see text] is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal [Formula: see text]-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal [Formula: see text]-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].