Journal of Homotopy and Related Structures最新文献

We describe various non-trivial examples that illustrate the approach to the “Teichmüller cocycle map” developed elsewhere in terms of crossed 2-fold extensions and generalizations thereof.

We show that the bigroupoid of semisimple symmetric Frobenius algebras over an algebraically closed field and the bigroupoid of Calabi–Yau categories are equivalent. To this end, we construct a trace on the category of finitely-generated representations of a symmetric, semisimple Frobenius algebra, given by the composite of the Frobenius form with the Hattori-Stallings trace.

We extend Massey products from cohomology to differential cohomology via stacks, organizing and generalizing existing constructions in Deligne cohomology. We study the properties and show how they are related to more classical Massey products in de Rham, singular, and Deligne cohomology. The setting and the algebraic machinery via stacks allow for computations and make the construction well-suited for applications. We illustrate with several examples from differential geometry and mathematical physics.

In his famous paper entitled “Operads and motives in deformation quantization”, Maxim Kontsevich constructed (in order to prove the formality of the little d-disks operad) a topological operad, which is called in the literature the Kontsevich operad, and which is denoted ({mathcal {K}}_d) in this paper. This operad has a nice structure: it is a multiplicative symmetric operad, that is, it comes with a morphism from the symmetric associative operad. There are many results in the literature regarding the formality of ({mathcal {K}}_d). It is well known (by Kontsevich) that ({mathcal {K}}_d) is formal over reals as a symmetric operad. It is also well known (independently by Syunji Moriya and the author) that ({mathcal {K}}_d) is formal as a multiplicative nonsymmetric operad. In this paper, we prove that the Kontsevich operad is formal over reals as a multiplicative symmetric operad, when (d ge 3).

This is the first in a series of papers constructing geometric models of twisted differential K-theory. In this paper we construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. By differential twists we will mean smooth U(1)-gerbes with connection, and we use twisted vector bundles with connection as cocycles. The model we construct satisfies the axioms of Kahle and Valentino, including functoriality, naturality of twists, and the hexagon diagram. This paper confirms a long-standing hypothetical idea that twisted vector bundles with connection define twisted differential K-theory.

Fix a symbol (underline{a}) in the mod-(ell ) Milnor K-theory of a field k, and a norm variety X for (underline{a}). We show that the ideal generated by (underline{a}) is the kernel of the K-theory map induced by (ksubset k(X)) and give generators for the annihilator of the ideal. When (ell =2), this was done by Orlov, Vishik and Voevodsky.

We show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the rational homology type of the input functor, whose layers are given by rational spectra with an action of O(n). By work of Greenlees and Shipley, we see that these layers are classified by torsion ({{mathrm{H}}}^*({{mathrm{B}}}SO(n))[O(n)/SO(n)])-modules.

For any Moore spectrum M and any homology theory ({{mathcal {H}}}_*), we associate a homology theory ({{mathcal {H}}}_*^M) which is related to ({{mathcal {H}}}_*) by a universal coefficient exact sequence of classical type. On the other hand the category of Moore spectra is not the category of ({mathbb {Z}})-modules, but it can be identified to a full subcategory of an abelian category ({{mathscr {D}}}). We prove that ({{mathcal {H}}}_*) can be lifted to a homology theory (widehat{{mathcal {H}}}_*) with values in ({{mathscr {D}}}) and we give a new universal coefficient exact sequence relating ({{mathcal {H}}}_*^M) and (widehat{{mathcal {H}}}_*) which is in general more precise than the classical one. We prove also a similar result for cohomology theories and we illustrate its convenience by computing the real K-theory of Moore spaces.

We show that the functor that takes a multicosimplicial object in a model category to its diagonal cosimplicial object is a right Quillen functor. This implies that the diagonal of a Reedy fibrant multicosimplicial object is a Reedy fibrant cosimplicial object, which has applications to the calculus of functors. We also show that, although the diagonal functor is a Quillen functor, it is not a Quillen equivalence for multicosimplicial spaces. We also discuss total objects and homotopy limits of multicosimplicial objects. We show that the total object of a multicosimplicial object is isomorphic to the total object of the diagonal, and that the diagonal embedding of the cosimplicial indexing category into the multicosimplicial indexing category is homotopy left cofinal, which implies that the homotopy limits are weakly equivalent if the multicosimplicial object is at least objectwise fibrant.

We develop a version of (G )-theory for an (mathbb F_1)-algebra (i.e., the (K )-theory of pointed G-sets for a pointed monoid G) and establish its first properties. We construct a Cartan assembly map to compare the Chu–Morava (K )-theory for finite pointed groups with our (G )-theory. We compute the (G )-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We introduce certain Loday–Whitehead groups over (mathbb F_1) that admit functorial maps into classical Whitehead groups under some reasonable hypotheses. We initiate a conjectural formalism using combinatorial Grayson operations to address the Equivariant Nishida Problem—it asks whether (mathbb {S}^G) admits operations that endow (oplus _npi _{2n}(mathbb {S}^G)) with a pre-(lambda )-ring structure, where G is a finite group and (mathbb {S}^G) is the G-fixed point spectrum of the equivariant sphere spectrum.