Journal of Homotopy and Related Structures最新文献

Voevodsky defined a motivic spectrum representing algebraic K-theory, and Panin, Pimenov, and R?ndigs described its ring structure up to homotopy. We construct a motivic symmetric spectrum with a strict ring structure. Then we show that these spectra are stably equivalent and that their ring structures are compatible up to homotopy.

In this paper, we study the extent to which Bousfield and finite localizations relative to a thick subcategory of equivariant finite spectra preserve various kinds of highly structured multiplications. Along the way, we describe some basic, useful results for analyzing categories of acyclics in equivariant spectra, and we show that Bousfield localization with respect to an ordinary spectrum (viewed as an equivariant spectrum with trivial action) always preserves equivariant commutative ring spectra.

In this article, we consider the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal subgroup of the orthogonal group of a non-degenerate quadratic space with a hyperbolic summand over a commutative ring, introduced by Roy. We prove a set of commutator relations among the elementary generators of the DSER elementary orthogonal group. As an application, we prove that this group is perfect and an action version of the Quillen’s local-global principle for this group is proved. This affirmatively answers a question of Rao in his Ph.D. thesis.

We revisit and improve Alex Heller’s results on the stabilization of derivators in Heller (J Pure Appl Algebra 115(2):113–130, 1997), recovering his results entirely. Along the way we give some details of the localization theory of derivators and prove some new results in that vein.

We give a new proof of the classical Dold–Thom theorem using factorization homology. Our method is direct and conceptual, avoiding the Eilenberg–Steenrod axioms entirely in favor of a more general geometric argument.

Tate provided an explicit way to kill a nontrivial homology class of a commutative differential graded algebra over a commutative noetherian ring R in Tate (Ill J Math 1:14–27, 1957). The goal of this article is to generalize his result to the case of GBV (Gerstenhaber–Batalin–Vilkovisky) algebras and, more generally, the descendant (L_infty )-algebras. More precisely, for a given GBV algebra ((mathcal {A}=oplus _{mge 0}mathcal {A}_m, delta , ell _2^delta )), we provide another explicit GBV algebra ((widetilde{mathcal {A}}=oplus _{mge 0}widetilde{mathcal {A}}_m, widetilde{delta }, ell _2^{widetilde{delta }})) such that its total homology is the same as the degree zero part of the homology (H_0(mathcal {A}, delta )) of the given GBV algebra ((mathcal {A}, delta , ell _2^delta )).

We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set X we associate a necklical set ({widehat{{varvec{Omega }}}}X) such that its geometric realization (|{widehat{{varvec{Omega }}}}X|), a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on |X| and the differential graded module of chains (C_*({widehat{{varvec{Omega }}}}X)) is a differential graded associative algebra generalizing Adams’ cobar construction.

In this paper we define a?family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a?standard d-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a?(Delta )-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an?odd-dimensional sphere. We develop the language to determine the homotopy type directly from the combinatorics of the indexing word. As added benefit of our investigation, we are able to emulate the Dunce hat phenomenon, and to obtain a?large family of both (Delta )-complexes, as well as simplicial complexes, which are contractible, but not collapsible.

We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown–Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson’s unstable splitting. As a consequence, we obtain an analog of Quillen’s theorem for Hodge filtered Brown–Peterson cohomology for complex manifolds.

In this paper we use topological tools to investigate the structure of the algebraic K-groups (K_4(R)) for (R=Z[i]) and (R=Z[rho ]) where (i := sqrt{-1}) and (rho := (1+sqrt{-3})/2). We exploit the close connection between homology groups of (mathrm {GL}_n(R)) for (nle 5) and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which (mathrm {GL}_n(R)) acts. Our main result is that (K_{4} ({mathbb {Z}}[i])) and (K_{4} ({mathbb {Z}}[rho ])) have no p-torsion for (pge 5).