Journal of Homotopy and Related Structures最新文献

Let Q denote the cyclic group of order two. Using the Tate diagram we compute the RO(Q)-graded coefficients of Eilenberg–MacLane Q-spectra and describe their structure as modules over the coefficients of the Eilenberg–MacLane spectrum of the Burnside Mackey functor. If the underlying Mackey functor is a Green functor, we also obtain the multiplicative structure on the RO(Q)-graded coefficients.

We study some properties of the coset poset associated with the family of subgroups of class (le 2) of a nilpotent group of class (le 3). We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is 2-Engel, and 2-connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the group of (4times 4) upper unitriangular matrices over (mathbb {F}_p), and for the Burnside groups of exponent 3.

Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, and Bohmann–Gerhardt–Høgenhaven–Shipley–Ziegenhagen developed a coBökstedt spectral sequence to compute the homology of (mathrm {coTHH}) for coalgebras over the sphere spectrum. We construct a relative coBökstedt spectral sequence to study (mathrm {coTHH}) of coalgebra spectra over any commutative ring spectrum R. Further, we use algebraic structures in this spectral sequence to complete some calculations of the homotopy groups of relative topological coHochschild homology.

We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic convergence theorems for the Real Johnson–Wilson theories (E_{mathbb {R}}(n)) hold only after Borel completion. We establish analogous results for the (C_{2^n})-equivariant Johnson–Wilson theories constructed by Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the available norms for an (N_infty )-algebra, and we determine which new norms appear. Finally, we explore generalizations of our results on smashing localizations in the context of a quasi-Galois extension of (E_infty )-rings.

The goal of this paper is to establish an equivalence of profinite completions of pro-spaces in model category theory and in (infty )-category theory. As an application, we show that the author’s comparison theorem for algebro-geometric objects in the setting of model categories recovers that of David Carchedi in the setting of (infty )-categories.

For a unital commutative Banach algebra A and its closed ideal I, we study the relative Čech cohomology of the pair ((mathrm {Max}(A),mathrm {Max}(A/I))) of maximal ideal spaces and show a relative version of the main theorem of Lupton et al. (Trans Amer Math Soc 361:267–296, 2009): (check{mathrm {H}}^{j}(mathrm {Max}(A),mathrm {Max}(A/I));{mathbb {Q}}) cong pi _{2n-j-1}(Lc_{n}(I))_{{mathbb {Q}}}) for (j < 2n-1), where (Lc_{n}(I)) refers to the space of last columns. We then study the rational cohomological dimension (mathrm {cdim}_{mathbb Q}mathrm {Max}(A)) for a unital commutative Banach algebra and prove an embedding theorem: if A is a unital commutative semi-simple regular Banach algebra such that (mathrm {Max}(A)) is metrizable and (mathrm {cdim}_{{mathbb {Q}}}mathrm {Max}(A) le m), then (i) the rational homotopy group (pi _{k}(GL_{n}(A))_{{mathbb {Q}}}) is stabilized if (n ge lceil (m+k+1)/2rceil ) and (ii) there exists a compact metrizable space (X_A) with (dim X_{A} le m) such that A is embedded into the commutative (C^*)-algebra (C(X_{A})) such that (pi _{k}(GL_{n}(C(X_{A})))) is rationally isomorphic to (pi _{k}(GL_{n}(A))) for each (kge 1) and (pi _{k}(GL_{n}(C(X_{A}))) is stabilized for (n ge lceil (m+k+1)/2 rceil ). The main technical ingredient is a modified version of a classical theorem of Davie (Proc Lond Math Soc 23:31–52, 1971).

Given an augmentation for a Legendrian surface in a 1-jet space, (Lambda subset J^1(M)), we explicitly construct an object, (mathcal {F} in mathbf {Sh}^bullet _{Lambda }(Mtimes mathbb {R}, mathbb {K})), of the (derived) category from Shende, Treumann and Zaslow (Invent Math 207(3), 1031–1133 (2017)) of constructible sheaves on (Mtimes mathbb {R}) with singular support determined by (Lambda ). In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in 1-jet spaces that, based on Rutherford and Sullivan (Cellular Legendrian contact homology for surfaces, Part I, arXiv:1608.02984.) Rutherford and Sullivan (Internat J Math 30(7):135, 2019) Rutherford and Sullivan (Internat J Math 30(7):111, 2019), is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of Shende, Treumann and Zaslow (Invent Math 207(3), 1031–1133 (2017)) for 1-dimensional Legendrian knots to obtain a combinatorial model for sheaves in (mathbf {Sh}^bullet _{Lambda }(Mtimes mathbb {R}, mathbb {K})) in the 2-dimensional case.

We explicitly construct and investigate a number of examples of ({mathbb {Z}}/p^r)-equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and p-divisible groups, as well as some other related examples.

We describe two ways to define higher order Toda brackets in a pointed simplicial model category ({mathcal {D}}): one is a recursive definition using model categorical constructions, and the second uses the associated simplicial enrichment. We show that these two definitions agree, by providing a third, diagrammatic, description of the Toda bracket, and explain how it serves as the obstruction to rectifying a certain homotopy-commutative diagram in ({mathcal {D}}).

We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or (infty )-groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean (infty )-topos.