Journal of Homotopy and Related Structures最新文献

This paper studies a notion of directed homology for preordered spaces, called the homology digraph. We show that the homology digraph is a directed homotopy invariant and establish variants of the main results of ordinary singular homology theory for the homology digraph. In particular, we prove a Künneth formula, which enables one to compute the homology digraph of a product of preordered spaces from the homology digraphs of the components.

By making use of Halperin’s local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau–de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout.

Let A be the Steenrod algebra over the finite field (k:= {mathbb {F}}_2) and G(q) be the general linear group of rank q over k. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, (textrm{Ext}^{q, *}_A(k, k),) for all homological degrees (q geqslant 0.) The Singer algebraic transfer of rank q,  formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of G(q) to (textrm{Ext}^{q, *}_A(k, k).) Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all (qgeqslant 4.) This paper establishes Singer’s conjecture for rank four in the generic degrees (n = 2^{s+t+1} +2^{s+1} - 3) whenever (tne 3) and (sgeqslant 1,) and (n = 2^{s+t} + 2^{s} - 2) whenever (tne 2,, 3,, 4) and (sgeqslant 1.) In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four.

We show that the category of non-counital conilpotent dg-coalgebras and the category of non-unital dg-algebras carry model structures compatible with their closed non-unital monoidal and closed non-unital module category structures respectively. Furthermore, we show that the Quillen equivalence between these two categories extends to a non-unital module category Quillen equivalence, i.e. providing an enriched form of Koszul duality.

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.