Journal of Homotopy and Related Structures最新文献

The aim of this paper is twofold. In the first part, we consider twisted Rota–Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an (L_infty )-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This leads to cohomology associated to a twisted Rota–Baxter operator. This cohomology can be seen as the Hochschild cohomology of a certain associative algebra with coefficients in a suitable bimodule. We study deformations of twisted Rota–Baxter operators by means of the above-defined cohomology. Application is given to Reynolds operators. In the second part, we consider NS-algebras of Leroux that are related to twisted Rota–Baxter operators in the same way dendriform algebras are related to Rota–Baxter operators. We define cohomology of NS-algebras using non-symmetric operads and study their deformations in terms of the cohomology.

We compute the Lusternik–Schnirelmann category and all the higher topological complexities of non-k-equal manifolds (M_d^{(k)}(n)) for certain values of d, k and n. This includes instances where (M_d^{(k)}(n)) is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring (H^*(M_d^{(k)}(n))) as described by Dobrinskaya and Turchin. A fine tuning comes from the use of obstruction theory techniques.

The goal of this short paper is to study the convergence of the Taylor tower of the identity functor in the context of operadic algebras in spectra. Specifically, we show that if A is a ((-1))-connected ({ mathcal {O} })-algebra with 0-connected ({ mathsf {TQ} })-homology spectrum ({ mathsf {TQ} }(A)), then there is a natural weak equivalence (P_infty ({ mathrm {id} })A simeq A^wedge _{ mathsf {TQ} }) between the limit of the Taylor tower of the identity functor evaluated on A and the ({ mathsf {TQ} })-completion of A. Since, in this context, the identity functor is only known to be 0-analytic, this result extends knowledge of the Taylor tower of the identity beyond its “radius of convergence.”

We introduce a construction that produces from each bialgebra H an operad (mathsf {Ass}_H) controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and hence does not fall within the reach of Koszul duality theory—so our work provides a new rich family of examples where an explicit minimal model of an operad can be obtained. As an application, we observe that if we take H to be the mod-2 Steenrod algebra ({mathscr {A}}), then this notion of an associative H-algebra coincides with the usual notion of an (mathscr {A})-algebra considered by homotopy theorists. This makes available to us an operad (mathsf {Ass}_{{mathscr {A}}}) along with its minimal model that controls the category of associative ({mathscr {A}})-algebras, and the notion of strong homotopy associative ({mathscr {A}})-algebras.

For a simply connected rationally elliptic CW-complex X, we show that the cohomology and the homotopy Euler–Poincaré characteristics are related to two new numerical invariants namely (eta _{X}) and (rho _{X}) which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of X.

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra (mathscr {A}^*). We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we improve upon a known connection between the graph theoretic interpretation of (mathscr {A}^*) and its structure as a Hopf algebra.

We introduce graded ({mathbb {E}}_{infty })-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective ({mathbb {N}})-graded ({mathbb {E}}_{infty })-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the (infty )-category of almost perfect quasi-coherent sheaves over a spectral projective scheme (text { {Proj}},(A)) associated to a connective ({mathbb {N}})-graded ({mathbb {E}}_{infty })-ring A can be described in terms of ({{mathbb {Z}}})-graded A-modules.

We compare different algebraic structures in twisted equivariant K-theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-theory, we prove a completion Theorem of Atiyah–Segal type for twisted equivariant K-theory. Using a universal coefficient theorem, we prove a cocompletion Theorem for twisted Borel K-homology for discrete groups.

We compute the homotopy groups of the (C_2) fixed points of equivariant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a ({mathrm {TMF}})-module, it is isomorphic to the tensor product of ({mathrm {TMF}}) with an explicit finite cell complex.