Journal of Homotopy and Related Structures最新文献

For (nin {2^t-3,2^t-2,2^t-1}) ((tge 3)) we study the cohomology algebra (H^*(widetilde{G}_{n,3};{mathbb {Z}}_2)) of the Grassmann manifold (widetilde{G}_{n,3}) of oriented 3-dimensional subspaces of ({mathbb {R}}^n.) A complete description of (H^*(widetilde{G}_{n,3};{mathbb {Z}}_2)) is given in the cases (n=2^t-3) and (n=2^t-2,) while in the case (n=2^t-1) we obtain a description complete up to a coefficient from ({mathbb {Z}}_2.)

Within the framework of Riehl–Shulman’s synthetic ((infty ,1))-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition à la Chevalley, Gray, Street, and Riehl–Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss discrete two-sided cartesian fibrations in this setting, corresponding to ((infty ,1))-distributors. The systematics of our definitions and results closely follows Riehl–Verity’s (infty )-cosmos theory, but formulated internally to Riehl–Shulman’s simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic ((infty ,1))-categories correspond to internal ((infty ,1))-categories implemented as Rezk objects in an arbitrary given ((infty ,1))-topos.

We define an (E_infty )-coalgebra structure on the chains of multisimplicial sets. Our primary focus is on the surjection chain complexes of McClure-Smith, for which we construct a zig-zag of complexity preserving quasi-isomorphisms of (E_infty )-coalgebras relating them to both the singular chains on configuration spaces and the Barratt–Eccles chain complexes.

Let ({mathcal {F}}) be a Morse–Bott foliation on the solid torus (T=S^1times D^2) into 2-tori parallel to the boundary and one singular central circle. Gluing two copies of T by some diffeomorphism between their boundaries, one gets a lens space (L_{p,q}) with a Morse–Bott foliation ({mathcal {F}}_{p,q}) obtained from ({mathcal {F}}) on each copy of T and thus consisting of two singular circles and parallel 2-tori. In the previous paper Khokliuk and Maksymenko (J Homotopy Relat Struct 18:313–356. https://doi.org/10.1007/s40062-023-00328-z, 2024) there were computed weak homotopy types of the groups ({mathcal {D}}^{lp}({mathcal {F}}_{p,q})) of leaf preserving (i.e. leaving invariant each leaf) diffeomorphisms of such foliations. In the present paper it is shown that the inclusion of these groups into the corresponding group ({mathcal {D}}^{fol}_{+}({mathcal {F}}_{p,q})) of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences.

In this paper, we study diffeological spaces as certain kinds of discrete simplicial presheaves on the site of cartesian spaces with the coverage of good open covers. The Čech model structure on simplicial presheaves provides us with a notion of (infty )-stack cohomology of a diffeological space with values in a diffeological abelian group A. We compare (infty )-stack cohomology of diffeological spaces with two existing notions of Čech cohomology for diffeological spaces in the literature Krepski et al. (Sheaves, principal bundles, and Čech cohomology for diffeological spaces. (2021). arxiv:2111 01032 [math.DG]), Iglesias-Zemmour (Čech-de-Rham Bicomplex in Diffeology (2020). http://math.huji.ac.il/piz/documents/CDRBCID.pdf). Finally, we prove that for a diffeological group G, that the nerve of the category of diffeological principal G-bundles is weak homotopy equivalent to the nerve of the category of G-principal (infty )-bundles on X, bridging the bundle theory of diffeology and higher topos theory.

We give a Matsumoto-type presentation of (K_2)-groups over rings of non-commutative Laurent polynomials, which is a non-commutative version of M. Tomie’s result for loop groups. Our main idea is induced by U. Rehmann’s approach in the case of division rings.

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.