Journal of Homotopy and Related Structures最新文献

In this paper, we consider Leibniz algebras with derivations. A pair consisting of a Leibniz algebra and a distinguished derivation is called a LeibDer pair. We define a cohomology theory for LeibDer pair with coefficients in a representation. We study central extensions of a LeibDer pair. In the next, we generalize the formal deformation theory to LeibDer pairs in which we deform both the Leibniz bracket and the distinguished derivation. It is governed by the cohomology of LeibDer pair with coefficients in itself. Finally, we consider homotopy derivations on sh Leibniz algebras and 2-derivations on Leibniz 2-algebras. The category of 2-term sh Leibniz algebras with homotopy derivations is equivalent to the category of Leibniz 2-algebras with 2-derivations.

Simons and Sullivan constructed a model of differential K-theory, and showed that the differential K-theory functor fits into a hexagon diagram. They asked whether, like the case of differential characters, this hexagon diagram uniquely determines the differential K-theory functor. This article provides a partial affirmative answer to their question: For any fixed compact manifold, the differential K-theory groups are uniquely determined by the Simons–Sullivan diagram up to an isomorphism compatible with the diagonal arrows of the hexagon diagram. We state a necessary and sufficient condition for an affirmative answer to the full question. This approach further yields an alternative proof of a weaker version of Simons and Sullivan’s results concerning axiomatization of differential characters. We further obtain a uniqueness result for generalised differential cohomology groups. The proofs here are based on a recent work of Pawar.

We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology, both known and new. This includes a recent generalization of Adams’s cobar-construction to the non-simply connected case, and a new algebraic model for the homotopy theory of connected topological spaces as an (infty )-category of discrete monoids.

We introduce the notion of cumulants as applied to linear maps between associative (or commutative) algebras that are not compatible with the algebraic product structure. These cumulants have a close relationship with (A_{infty }) and (C_{infty }) morphisms, which are the classical homotopical tools for analyzing deformations of algebraically compatible linear maps. We look at these two different perspectives to understand how infinity-morphisms might inform our understanding of cumulants. We show that in the presence of an (A_{infty }) or (C_{infty }) morphism, the relevant cumulants are strongly homotopic to zero.

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke–Turner–Willerton to include open strings. At the same time, it generalises work of Moore–Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed–Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda–Pfeiffer.

We examine the relation between the gauge groups of (mathrm {SU}(n))- and (mathrm {PU}(n))-bundles over (S^{2i}), with (2le ile n), particularly when n is a prime. As special cases, for (mathrm {PU}(5))-bundles over (S^4), we show that there is a rational or p-local equivalence (mathcal {G}_{2,k}simeq _{(p)}mathcal {G}_{2,l}) for any prime p if, and only if, ((120,k)=(120,l)), while for (mathrm {PU}(3))-bundles over (S^6) there is an integral equivalence (mathcal {G}_{3,k}simeq mathcal {G}_{3,l}) if, and only if, ((120,k)=(120,l)).

In the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing the algebraic K-theory of X(n) using trace methods, which approximates the algebraic K-theory of the sphere spectrum in a precise sense. We solve the homotopy limit problem for topological Hochschild homology of T(n) under the assumption that the canonical map (T(n)rightarrow BP) of homotopy commutative ring spectra can be rigidified to map of (E_2) ring spectra. We show that the obstruction to our assumption holding can be described in terms of an explicit class in an Atiyah-Hirzebruch spectral sequence.

The purpose of this article is to record the computation of the homotopy groups of 3-local (mathrm {tmf}) via the Adams spectral sequence.

We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to a c-group and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.

We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras by mixing the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also introduce a formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations.