Mathematics

For fixed j and w, we study the j-th homology of the configuration space of n labeled disks of width 1 in an infinite strip of width w. As n grows, the homology groups grow exponentially in rank, suggesting a generalized representation stability as defined by Church--Ellenberg--Farb and Ramos. We prove this generalized representation stability for the strip of width 2, leaving open the case of w > 2. We also prove it for the configuration space of n labeled points in the line, of which no k are equal.

We examine the rationality conjecture which states that (a) the formal power series $\sum_{r\ge 1} \tc_{r+1}(X)\cdot x^r$ represents a rational function of x with a single pole of order 2 at x=1 and (b) the leading coefficient of the pole equals $\cat(X)$. Here X is a finite CW-complex and for r≥2 the symbol $\tc_r(X)$ denotes its r -th sequential topological complexity. We analyse an example (violating the Ganea conjecture) and conclude that part (b) of the rationality conjecture is false in general. Besides, we establish a cohomological version of the rationality conjecture.

In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we states our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application.

We interpret a construction of geometric realisation by [Besser], [Grayson], and [Drinfeld] of a simplicial set as constructing a space of maps from the interval to a simplicial set, in a certain formal sense, reminiscent of the Skorokhod space of semi-continuous functions; in particular, we show the geometric realisation functor factors through an endofunctor of a certain category. Our interpretation clarifies the explanation of [Drinfeld] "why geometric realization commutes with Cartesian products and why the geometric realization of a simplicial set [...] is equipped with an action of the group of orientation preserving homeomorphisms of the segment [0,1]".

Ginzburg algebras associated to triangulated surfaces provide a means to categorify the cluster algebras of these surfaces. As shown by Ivan Smith, the finite derived category of such a Ginzburg algebra can be embedded into the Fukaya category of the total space of a Lefschetz fibration over the surface. Inspired by this perspective we provide a description of the full derived category in terms of a perverse schober. The main novelty is a gluing formalism describing the Ginzburg algebra as a colimit of certain local Ginzburg algebras associated to discs. As a first application we give a new proof of the derived invariance of these Ginzburg algebras under flips of an edge of the triangulation. Finally, we note that the perverse schober as well as the resulting gluing construction can also be defined over the sphere spectrum.

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth-Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behavior is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we compute the first page of the integral Bousfield--Kan homotopy spectral sequence of the tower of fibrations given by the Taylor tower of the embedding functor associated to the space of long knots. Based on the methods in [Con08], we give a combinatorial interpretation of the differentials d 1 mapping into the diagonal terms, by introducing the notion of (i,n) -marked unitrivalent graphs.

We define a new family of graph invariants, studying the topology of the moduli space of their geometric realizations in Euclidean spaces, using a limiting procedure reminiscent of Floer homology. Given a labeled graph G on n vertices and d≥1 , W G,d ⊆ R d×n denotes the space of nondegenerate realizations of G in R d .The set W G,d might not be connected, even when it is nonempty, and we refer to its connected components as rigid isotopy classes of G in R d . We study the topology of these rigid isotopy classes. First, regarding the connectivity of W G,d , we generalize a result of Maehara that W G,d is nonempty for d≥n to show that W G,d is k -connected for d≥n+k+1 , and so W G,∞ is always contractible. While π k ( W G,d )=0 for G , k fixed and d large enough, we also prove that, in spite of this, when d→∞ the structure of the nonvanishing homology of W G,d exhibits a stabilization phenomenon: it consists of (n−1) equally spaced clusters whose shape does not depend on d , for d large enough. This leads to the definition of a family of graph invariants, capturing this structure. For instance, the sum of the Betti numbers of W G,d does not depend on d , for d large enough; we call this number the Floer number of the graph G . Finally, we give asymptotic estimates on the number of rigid isotopy classes of R d --geometric graphs on n vertices for d fixed and n tending to infinity. When d=1 we show that asymptotically as n→∞ each isomorphism class corresponds to a constant number of rigid isotopy classes, on average. For d>1 we prove a similar statement at the logarithmic scale.

We show that the pretensor and tensor products of simplicial sets with marking are compatible with the homotopy theory of saturated N -complicial sets (which are a proposed model of (∞,N) -categories), in the form of a Quillen bifunctor and a homotopical bifunctor, respectively.

We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bifunctor with respect to the bicategorical model category of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings.