Journal of Homotopy and Related Structures最新文献

M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces S of an Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing S. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing S. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex. In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann’s CS sets. In terms of King’s paper, this means that polyhedral dimension is a “reasonable” dimension. The proof uses a Mayer-Vietoris argument which needs an adapted subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them.

In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted K-theory of connected, simply-connected and simple compact Lie groups G, in both equivariant and nonequivariant settings.

We examine the homotopy types of Vietoris–Rips complexes on certain finite metric spaces at scale 2. We consider the collections of subsets of ([m]={1, 2, ldots , m}) equipped with symmetric difference metric d, specifically, ({mathcal {F}}^m_n), ({mathcal {F}}_n^mcup {mathcal {F}}^m_{n+1}), ({mathcal {F}}_n^mcup {mathcal {F}}^m_{n+2}), and ({mathcal {F}}_{preceq A}^m). Here ({mathcal {F}}^m_n) is the collection of size n subsets of [m] and ({mathcal {F}}_{preceq A}^m) is the collection of subsets (preceq A) where (preceq ) is a total order on the collections of subsets of [m] and (Asubseteq [m]) (see the definition of (preceq ) in Sect. 1). We prove that the Vietoris–Rips complexes ({{mathcal {V}}}{{mathcal {R}}}({mathcal {F}}^m_n, 2)) and ({{mathcal {V}}}{{mathcal {R}}}({mathcal {F}}_n^mcup {mathcal {F}}^m_{n+1}, 2)) are either contractible or homotopy equivalent to a wedge sum of (S^2)’s; also, the complexes ({{mathcal {V}}}{{mathcal {R}}}({mathcal {F}}_n^mcup {mathcal {F}}^m_{n+2}, 2)) and ({{mathcal {V}}}{{mathcal {R}}}({mathcal {F}}_{preceq A}^m, 2)) are either contractible or homotopy equivalent to a wedge sum of (S^3)’s. We provide inductive formulae for these homotopy types extending the result of Barmak about the independence complexes of Kneser graphs KG(_{2, k}) and the result of Adamaszek and Adams about Vietoris–Rips complexes of hypercube graphs with scale 2.

In this paper, we study nonabelian extensions of associative algebras using associative 2-algebra homomorphisms. First we construct an associative 2-algebra using the bimultipliers of an associative algebra. Then we classify nonabelian extensions of associative algebras using associative 2-algebra homomorphisms. Finally we analyze the relation between nonabelian extensions of associative algebras and nonabelian extensions of the corresponding commutator Lie algebras.

In this article, we show that for a quasicompact scheme X and (n>0,) the n-th K-group (K_{n}(X)) is a (lambda )-module over a (lambda )-ring (K_{0}(X)) in the sense of Hesselholt.

In (Xu, J Pure Appl Algebra 212:2555–2569, 2008), a LHS-spectral sequence for target regular extensions of small categories is constructed. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). As a special case of these LHS-spectral sequences, we obtain three different versions of Słomińska’s spectral sequence for the cohomology of regular EI-categories. We show that many well-known spectral sequences related to the homology decompositions of finite groups, centric linking systems, and the orbit category of fusion systems can be obtained as the LHS-spectral sequence of an extension.

In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over ({mathbb {C}}). In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic Morava K-theories and the preimages of the thick ideals in the stable homotopy category under Betti realization.

We compute the homotopy Mackey functors of the (KU_G)-local equivariant sphere spectrum when G is a finite q-group for an odd prime q, building on the degree zero case due to Bonventre and the third and fifth authors.

Historically, it was known by the work of Artin and Mazur that the (ell )-adic homotopy type of a smooth complex variety with good reduction mod p can be recovered from the reduction mod p, where (ell ) is not p. This short note removes this last constraint, with an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor of Mandell, we see that the étale comparison theorem in the prismatic theory reproduces the p-adic homotopy type for a smooth complex variety with good reduction mod p.

Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category (Delta ) to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in (Delta ) to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in (Delta ), which we characterize completely, along with several other classes of squares in (Delta ). Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example.