We classify the localizing tensor ideals of the integral stable module category for any finite group $G$. This results in a generic classification of $mathbb{Z}[G]$-lattices of finite and infinite rank and globalizes the modular case established in c
elebrated work of Benson, Iyengar, and Krause. Further consequences include a verification of the generalized telescope conjecture in this context, a tensor product formula for integral cohomological support, as well as a generalization of Quillens stratification theorem for group cohomology. Our proof makes use of novel descent techniques for stratification in tensor-triangular geometry that are of independent interest.
A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study vario
Rudyaks conjecture states that cat$(M) geq$ cat$(N)$ given a degree one map $f:M to N$ between closed manifolds. We generalize this conjecture to sectional category, and follow the methodology of [5] to get the following result: Given a normal map of
degree one $f:M to N$ between smooth closed manifolds, fibrations $p^M:E^M to M$ and $p^N:E^N to N$, and lift $overline{f}$ of $f$ with respect to $p^M$ and $p^N$, i.e., $fp^M = overline{f} p^N$; then if $f$ has no surgery obstructions and $N$ satisfies the inequality $5 leq dim N leq 2r$ secat$(p^N) - 3$ (where the fiber of $p^N$ is $(r-2)$-connected for some $r geq 1$), then secat$(p^M) geq$secat$(p^N)$. Finally, we apply this result to the case of higher topological complexity when $N$ is simply connected.
We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order $[n]$, we enumerate all model structures, exhibiting a rich combinatorial structure e
ncoded by Shapiros Catalan triangle. This is an application of previous work of the authors on the theory of $N_infty$-operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of $[n]$.
The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Cont
ractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be ${mathcal I}$-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of ${mathcal I}$-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ, as well as a miminal counterexample to an erroneous claim of Ivashchenko from the literature. We also relate these classes of graphs to the notion of $k$-dismantlable graphs and $k$-collapsible complexes, and discuss some open questions.
In a 2005 paper, Casacuberta, Scevenels and Smith construct a homotopy idempotent functor $E$ on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map $f$ is independent of the ZFC ax
ioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe $mathcal{U}$. When specialized to an appropriate family, this produces a localization which when interpreted in the $infty$-topos of spaces agrees with the localization corresponding to $E$. Our approach generalizes the approach of [CSS] in two ways. First, by working in homotopy type theory, our construction can be interpreted in any $infty$-topos. Second, while the local objects produced by [CSS] are always 1-types, our construction can produce $n$-types, for any $n$. This is new, even in the $infty$-topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about small types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice which implies that sets cover and that the law of excluded middle holds.
In this paper we investigate a topological characterization of the Runge theorem in the Clifford algebra $ mathbb{R}_3$ via the description of the homology groups of axially symmetric open subsets of the quadratic cone in $mathbb{R}_3$.
This paper is devoted to the study of Keller admissible triples. We prove that a Keller admissible triple induces an isomorphism of Gerstenhaber algebras between Hochschild cohomologies of the direct-sum type for dg algebras. As an application, we gi
ve an alternative proof of the Kontsevich-Duflo theorem for finite-dimensional Lie algebras.
We recall several categories of graphs which are useful for describing homotopy-cohere
We present a local combinatorial formula for Euler class of $n$-dimensional PL spherical fiber bundle as a rational number $e_{it CH}$ associated to chain of $n+1$ abstract subdivisions of abstract $n$-spherical PL cell complexes. The number $e_{it C
H}$ is combinatorial (or matrix) Hodge theory twisting cochain in Guy Hirshs homology model of the bundle associated with PL combinatorics of the bundle.
