Journal of Homotopy and Related Structures最新文献

Let G be discrete group and (mathcal F) be a collection of subgroups of G. We show that there exists a left induced model structure on the category of right G-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on H-orbits for all H in (mathcal F). This gives a model categorical criterion for maps that induce weak equivalences on H-orbits to be weak equivalences in the (mathcal F)-model structure.

Given a fusion system ({mathcal {F}}) defined on a p-group S, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize ({mathcal {F}}). We study these models when ({mathcal {F}}) is a fusion system of a finite group G and prove a theorem which relates the cohomology of an infinite group model (pi ) to the cohomology of the group G. We show that for the groups GL(n,?2), where (nge 5), the cohomology of the infinite group obtained using the Robinson model is different than the cohomology of the fusion system. We also discuss the signalizer functors (Prightarrow Theta (P)) for infinite group models and obtain a long exact sequence for calculating the cohomology of a centric linking system with twisted coefficients.

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid (Pi _1(X)) has well-defined dense products if and only if X admits a generalized universal covering space.

We formulate the concept of minimal fibration in the context of fibrations in the model category ({mathbf {S}}^{mathcal {C}}) of ({mathcal {C}})-diagrams of simplicial sets, for a small index category ({mathcal {C}}). When ({mathcal {C}}) is an EI-category satisfying some mild finiteness restrictions, we show that every fibration of ({mathcal {C}})-diagrams admits a well-behaved minimal model. As a consequence, we establish a classification theorem for fibrations in ({mathbf {S}}^{mathcal {C}}) over a constant diagram, generalizing the classification theorem of Barratt, Gugenheim, and Moore for simplicial fibrations (Barratt?et?al. in Am J Math 81:639–657, 1959).

We exploit the equivalence between t-structures and normal torsion theories on a stable (infty )-category to show how a few classical topics in the theory of triangulated categories, i.e., the characterization of bounded t-structures in terms of their hearts, their associated cohomology functors, semiorthogonal decompositions, and the theory of tiltings, as well as the more recent notion of Bridgeland’s slicings, are all particular instances of a single construction, namely, the tower of a morphism associated with a J-slicing of a stable (infty )-category , where J is a totally ordered set equipped with a monotone (mathbb {Z})-action.

We use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, with a complete classification finally appearing in the 1990s. In this paper we give a different approach to the classification, using techniques that are more accessible to algebraic topologists as well as a new invariant (which we call the double-Dickson invariant) for distinguishing the “hard” cases.

We define a comonad cohomology of track categories, and show that it is related via a long exact sequence to the corresponding (({mathcal {S}}!,!mathcal {O}))-cohomology. Under mild hypotheses, the comonad cohomology coincides, up to reindexing, with the (({mathcal {S}}!,!mathcal {O}))-cohomology, yielding an algebraic description of the latter. We also specialize to the case where the track category is a 2-groupoid.

We introduce twisted matrix factorizations for quantum complete intersections of codimension two. For such an algebra, we show that in a given dimension, almost all the indecomposable modules with bounded minimal projective resolutions correspond to such factorizations.

We discuss two categorical characterizations of the class of acyclic maps between spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks along (pi _0)-surjective maps define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined by the class of acyclic maps, and discuss the content of the generalized Blakers–Massey theorem for this modality.

Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for (G=({mathbb {Z}}/2)^n) was completely calculated by Bruner and Greenlees (The connective K-theory of finite groups, 2003). In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to (p>2) prime. We also identify the resulting spectra, which are products of Eilenberg–Mac Lane spectra, and finitely many finite Postnikov towers. For (p=2), we also reconcile our answer completely with the result of [2], which is in a different form, and hence the comparison involves some non-trivial combinatorics.