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Labels instead of coefficients: a label bracket which dominates the Jones polynomial, the Kuperberg bracket, and the normalised arrow polynomialJul 15 2019In the present paper we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial, the Kuperberg bracket, and the normalized arrow polynomial.
Depth and detection for Noetherian unstable algebrasJul 15 2019For a connected Noetherian unstable algebra $R$ over the mod $p$ Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of $R$, originally proved when $R$ is the mod $p$ cohomology ring of a finite group. This recovers the ... More
Metric Thickenings, Borsuk-Ulam Theorems, and OrbitopesJul 14 2019Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle ... More
Stability of Bott--Samelson Classes in Algebraic CobordismJul 13 2019In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by ... More
Infinity-enhancing Leibniz algebrasJul 12 2019We show that the recent formulation of tensor hierarchies in terms of infinity-enhanced Leibniz algebras is a consequence of the differential graded Lie algebra structure already used in this context by constructions from a Leibniz algebra or a tensor ... More
Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifoldsJul 12 2019The Torelli group $\mathcal T(X)$ of a closed smooth manifold $X$ is the subgroup of the mapping class group $\pi_0(\mathrm{Diff}^+(X))$ consisting of elements which act trivially on the integral cohomology of $X$. In this note we give counterexamples ... More
Topological realizations of groups in Alexandroff spacesJul 12 2019Given a group $G$, we provide a constructive method to get infinitely many (non-homotopy-equivalent) Alexandroff spaces, such that the group of autohomeomorphisms, the group of homotopy classes of self-homotopy equivalences and the pointed version are ... More
Quadratic Algebras arising from Hopf operads generated by a single elementJul 12 2019The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated ... More
Nearly Frobenius AlgebrasJul 11 2019In this introductory paper we study nearly Frobenius algebras which are generalizations of the concept of a Frobenius algebra which appear naturally in topology: nearly Frobenius algebras have no traces (co-units). We survey the most basic foundational ... More
The constructive Kan-Quillen model structure: two new proofsJul 11 2019We present two new proofs of Simon Henry's result that the category of simplicial sets admits a constructive counterpart of the classical Kan-Quillen model structure. Our proofs are entirely self-contained and avoid complex combinatorial arguments on ... More
Bernstein--Sato Varieties, $\mathscr{D}_{X}[S]F^{S}$, the Map $\nabla_{A}$, and Cohomology Support LociJul 11 2019Given a complex germ $f$ near the point $\mathfrak{x}$ of the complex manifold $X$, equipped with a factorization $f = f_{1} \cdots f_{r}$, we consider the $\mathscr{D}_{X,\mathfrak{x}}[s_{1}, \dots, s_{r}]$-module generated by $ F^{S} := f_{1}^{s_{1}} ... More
Distributing Persistent Homology via Spectral SequencesJul 11 2019We set up the theory for a distributive algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of using them. Our ... More
Homotopy invariance of convolution productsJul 11 2019The purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of simplicial sets ... More
Chromatic (co)homology of finite general linear groupsJul 11 2019We study the Morava $E$-theory (at a prime $p$) of $BGL_d(F)$, where $F$ is a finite field with $|F|=1\pmod{p}$. Taking all $d$ together, we obtain a structure with two products $\times$ and $\bullet$. We prove that it is a polynomial ring under $\times$, ... More
Computing Minimal Persistent Cycles: Polynomial and Hard CasesJul 10 2019Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in the purely topological persistence diagrams (also termed as barcodes). In our earlier work, we showed that computing minimal ... More
The cohomology rings of smooth toric varieties and quotients of moment-angle complexesJul 10 2019Partial quotients of moment-angle complexes are topological analogues of smooth, not necessarily compact toric varieties. In 1998, Buchstaber and Panov proposed a formula for the cohomology ring of such a partial quotient in terms of a torsion product ... More
Homotopy Gerstenhaber formality of Davis-Januszkiewicz spacesJul 10 2019A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. We prove that the normalized singular cochain algebra of a Davis-Januszkiewicz space is formal ... More
Homotopy Gerstenhaber algebras are strongly homotopy commutativeJul 10 2019We show that any homotopy Gerstenhaber algebra is canonically a strongly homotopy commutative (shc) algebra in the sense of Stasheff-Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to ... More
The cohomology rings of homogeneous spacesJul 10 2019Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in this case the ... More
A commutative model for PL compactly supported cohomology in characteristic zeroJul 09 2019We show that that classical rational homotopy theory in the sense of Sullivan [6] can be extended compactly supported setting. This presents a simplicial version of the compactly supported de Rham complex in characteristic zero, and proving that it models ... More
Split Hopf algebras, quasi-shuffle algebras, and the cohomology of Omega Sigma XJul 09 2019Let A and B be two connected graded commutative k-algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only if A and B are ... More
Homotopy-coherent algebra via Segal conditionsJul 09 2019Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an "algebraic pattern", by which we mean an $\infty$-category equipped with a factorization system and a collection of "elementary" objects; examples ... More
Homology of posets with functor coefficients and its relation to Khovanov homology of knotsJul 09 2019We study homology groups of posets with functor coefficients and apply our results to give a novel approach to study Khovanov homology of knots and related homology theories.
Fibrations between finite topological spacesJul 09 2019We study Hurewicz fibrations between finite T$_0$--spaces from a combinatorial viewpoint and give strong conditions that a continuous map between finite T$_0$--spaces must satisfy in order to be a Hurewicz fibration. We also show that there exists a strong ... More
Generalized bornological coarse spaces and coarse motivic spectraJul 09 2019We generalize the notion of a bornology by omitting the condition that a one-point-subset is bounded and obtain a complete and co-complete generalization of the category of bornological coarse spaces. Then we imitate the construction of motivic coarse ... More
Arithmetic topology of 4-manifoldsJul 08 2019We construct a functor from the smooth 4-dimensional manifolds to the hyper-algebraic number fields, i.e. fields with a non-commutative multiplication. It is proved that that the simply connected 4-manifolds correspond to the abelian extensions. As a ... More
$L'$-localization in an $\infty$-toposJul 08 2019We prove that, given any reflective subfibration $L_\bullet$ on an $\infty$-topos $\mathcal{E}$, there exists a reflective subfibration $L'_\bullet$ on $\mathcal{E}$ whose local maps are the $L$-separated maps, that is, the maps whose diagonals are $L$-local. ... More
Localization theory in an $\infty$-toposJul 08 2019We develop the theory of reflective subfibrations on an $\infty$-topos $\mathcal{E}$. A reflective subfibration $L_\bullet$ on $\mathcal{E}$ is a pullback-compatible assignment of a reflective subcategory $\mathcal{D}_X\subseteq \mathcal{E}{/X}$, for ... More
A new tool to study the fixed point property of finite posetsJul 08 2019We develop a novel tool to study the fixed point property of finite posets using a topological approach. Our tool is a construction which turns out to induce an endofunctor of the homotopy category of finite $T_0$--spaces. We study many properties of ... More
Classification of fiber bundles over Alexandroff spaces with T$_0$ fiberJul 08 2019We introduce a variant of the Grothendieck construction by means of which we give a classification theorem for fiber bundles over Alexandroff spaces with T$_0$ fiber. As a corollary we obtain that any fiber bundle with T$_0$ fiber over a simply-connected ... More
Coarse Homotopy on metric Spaces and their CoronaJul 08 2019This paper discusses properties of the Higson corona by means of a quotient on coarse ultrafilters on a proper metric space. We use this description to show that the corona functor is faithful. This study provides a K\"unneth formula for twisted coarse ... More
Bökstedt periodicity and quotients of DVRsJul 08 2019In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for B\"okstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the ... More
Cyclic structures and broken cyclesJul 07 2019We introduce a new way to encode semicyclic structures using a stack of broken cycles. (We also prove an analogue for paracyclic structures.) This was motivated not only by higher algebra but also by Fukaya-categorical considerations. We also openly speculate ... More
The homotopy types of $U(n)$-gauge groups over lens spacesJul 05 2019We analyse the homotopy types of gauge groups for principal $U(n)$-bundles over lens spaces.
An Introduction to Higher Categorical AlgebraJul 05 2019This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable $\infty$-categories, such as the ... More
Cyclic homology for bornological coarse spacesJul 05 2019We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and $\mathcal{X}HC_{k}^G$ from the category of equivariant bornological coarse ... More
Regular coverings and fundamental groupoids of Alexandroff spacesJul 04 2019We summarize several results about the regular coverings and the fundamental groupoids of Alexandroff spaces. In particular, we show that the fundamental groupoid of an Alexandroff space $X$ is naturally isomorphic to the localization, at its set of morphisms, ... More
Degree theory for orbifoldsJul 04 2019In [3] Borzellino and Brunsden started to develop an elementary differential topology theory for orbifolds. In this paper we carry on their project by defining a mapping degree for proper maps between orbifolds, which counts preimages of regular values ... More
Quillen's conjecture for groups of p-rank 3Jul 03 2019Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ for any group $G$ ... More
A homotopy coherent cellular nerve for bicategoriesJul 03 2019The subject of this paper is a nerve construction for bicategories introduced by Leinster, which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over Joyal's category $\Theta_2$. ... More
Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cyclesJul 03 2019We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic ... More
Simplicial complexity of surface groups and systolic areaJul 02 2019The simplicial complexity is an invariant for finitely presentable groups that was recently introduced by Babenko, Balacheff and Bulteau to study systolic area. The simplicial complexity $\kappa(G)$ was proved to be a good approximation of the systolic ... More
The Bredon-Landweber region in $C_2$-equivariant stable homotopy groupsJul 02 2019We use the $C_2$-equivariant Adams spectral sequence to compute part of the $C_2$-equivariant stable homotopy groups $\pi^{C_2}_{n,n}$. This allows us to recover results of Bredon and Landweber on the image of the geometric fixed-points map from the equivariant ... More
Left properness of multipointed d-spacesJul 02 2019We prove that the q-model structure and the m-model structure of multipointed $d$-spaces are left proper. We also use the techniques developed in this paper to consider the limit case in which all lengths of execution paths are equal to $0$. We then obtain, ... More
Lifting in compact covering spaces for fractional Sobolev mappingsJul 02 2019Let $\pi : \widetilde{\mathcal{N}} \to \mathcal{N}$ be a Riemannian covering, with $\mathcal{N}$, $\widetilde{\mathcal{N}}$ smooth compact connected Riemannian manifolds. If $\mathcal{M}$ is an $m$-dimensional compact simply-connected Riemannian manifold, ... More
Equivariant instanton homologyJul 01 2019We define four versions of equivariant instanton Floer homology ($I^+, I^-, I^\infty$ and $\widetilde I$) for a class of 3-manifolds and $SO(3)$-bundles over them including all rational homology spheres. These versions are analogous to the four flavors ... More
Biased permutative equivariant categoriesJul 01 2019For a finite group G, we introduce the complete suboperad $Q_G$ of the categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ G-operad (i.e., it is $N_\infty$ ... More
Improved hardness for H-colourings of G-colourable graphsJul 01 2019We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring $k$-colourable graphs with $\binom{k}{\lfloor k/2\rfloor}-1$ ... More
Improved hardness for H-colourings of G-colourable graphsJul 01 2019Jul 05 2019We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring $k$-colourable graphs with $\binom{k}{\lfloor k/2\rfloor}-1$ ... More
Cardinal-indexed classifying spaces for families of subgroups of any topological groupJul 01 2019Any $G$-space isovariantly or approximately covered by tubes is the pullback of a classifying space indexed by the orbit types of the tubes and the cardinality of the cover.
Random Simplicial Complexes in the Medial RegimeJul 01 2019Jul 02 2019We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters $p_\sigma$ approach neither $0$ nor $1$. We show that nontrivial Betti numbers of typical ... More
The two digital homology theoriesJun 30 2019In this paper we prove results relating to four homology theories developed in the topology of digital images: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by Lee, a cubical ... More
Framed motivic $Γ$-spacesJun 30 2019We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines $\Gamma$-spaces and framed correspondences into ... More
Lectures on Factorization Homology, Infinity-Categories, and Topological Field TheoriesJun 28 2019These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are not homotopy theorists.
Coextension of scalars in operad theoryJun 28 2019The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functor to admit a right adjoint. The condition is a factorization axiom which ... More
Comparison of spaces associated to DGLA via higher holonomyJun 27 2019Fof a nilpotent differential graded Lie algebra whose components vanish in degrees below -1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of differential forms with values in the Lie algebra introduced ... More
The homotopy groups of the η-periodic motivic sphere spectrumJun 27 2019We compute the homotopy groups of the {\eta}-periodic motivic sphere spectrum over a finite-dimensional field k such that (a) k has odd characteristic, (b) k has 2-cohomological dimension at most 2, or (c) k contains a square root of -1. We also study ... More
Burnside rings for Real $2$-representation theory: The linear theoryJun 26 2019Jul 02 2019This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction ... More
Burnside rings for Real $2$-representation theory: The linear theoryJun 26 2019This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the usual (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction ... More
Morse theory without nondegeneracyJun 26 2019We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any nondegeneracy assumptions except that the critical locus must have only finitely many connected components.
Towards a taxonomy of atlases and of morphisms between themJun 25 2019Manifolds and fiber bundles, while superficially different, have strong parallels; in particular, they are both defined in terms of equivalence classes of atlases or in terms of maximal atlases, with the atlases treated as mere adjuncts. This paper presents ... More
Higher Segal spaces via higher excisionJun 25 2019Jun 26 2019We show that the various higher Segal conditions of Dyckerhoff and Kapranov can all be characterized in purely categorical terms by higher excision conditions (in the spirit of Goodwillie-Weiss manifold calculus) on the simplex category $\Delta$ and the ... More
Higher Segal spaces via higher excisionJun 25 2019We show that the various higher Segal conditions of Dyckerhoff and Kapranov can all be characterized in purely categorical terms by higher excision conditions (in the spirit of Goodwillie-Weiss manifold calculus) on the simplex category $\Delta$ and the ... More
Computing persistent homology of directed flag complexesJun 25 2019We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation ... More
A co-reflection of cubical sets into simplicial sets with applications to model structuresJun 21 2019We show that the category of simplicial sets is a co-reflective subcategory of the category of cubical sets with connections, with the inclusion given by a version of the straightening functor. We show that using the co-reflector, one can transfer any ... More
Simplex2Vec embeddings for community detection in simplicial complexesJun 21 2019Topological representations are rapidly becoming a popular way to capture and encode higher-order interactions in complex systems. They have found applications in disciplines as different as cancer genomics, brain function, and computational social science, ... More
Inessential directed maps and directed homotopy equivalencesJun 21 2019A directed space is a topological space $X$ together with a subspace $\vec{P}(X)\subset X^I$ of \emph{directed} paths on $X$. A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology of the associated ... More
Endomorphism operads of functorsJun 21 2019We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been ... More
Endomorphism operads of functorsJun 21 2019Jul 03 2019We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been ... More
Connectivity-Optimized Representation Learning via Persistent HomologyJun 21 2019We study the problem of learning representations with controllable connectivity properties. This is beneficial in situations when the imposed structure can be leveraged upstream. In particular, we control the connectivity of an autoencoder's latent space ... More
Explicit modular forms from the divided beta familyJun 21 2019We compute modular forms known to arise from the order 5 generators of the 5-local Adams-Novikov spectral sequence 2-line, generalizing and contextualizing previous computations of M. Behrens and G. Laures. We exhibit analogous computations at other primes ... More
Minimal resolutions of monomial idealsJun 20 2019An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional generalizations of combined ... More
Polyiamonds Attaining Extremal Topological PropertiesJun 20 2019We consider two optimization questions with respect to polyiamonds. What is the maximum number of holes that a polyiamond with $n$ tiles can enclose, and what is the minimum number of tiles required to construct a polyiamond with $h$ holes? These numbers ... More
An alternative approach to the calculation of fundamental groups based on labeled natural deductionJun 19 2019In this work, we use a labelled deduction system based on the concept of computational paths (sequence of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these ... More
A Topological Application of Labelled Natural DeductionJun 19 2019We use a labelled deduction system based on the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational ... More
Equivariant higher twisted K-theory of SU(n) for exponential functor twistsJun 19 2019We prove that each exponential functor on the category of finite-dimensional complex inner product spaces and isomorphisms gives rise to an equivariant higher (ie. non-classical) twist of $K$-theory over $G=SU(n)$. This twist is represented by a Fell ... More
Jaccard Filtration and Stable Paths in the MapperJun 19 2019The contributions of this paper are two-fold. We define a new filtration called the cover filtration built from a single cover based on a generalized Jaccard distance. We provide stability results for the cover filtration and show how the construction ... More
The Fulton Mac Pherson operad and the W-constructionJun 18 2019In this short note we explain in detail the construction of a $O(n)$-equivariant isomorphism of topological operads $F_n \cong WF_n$ , where $F_n$ is the Fulton Mac Pherson operad and $W$ is the Boardman-Vogt construction
A cell decomposition of the Fulton MacPherson operadJun 18 2019We construct a regular cellular decomposition of the Fulton MacPherson operad $FM_2$ that is compatible with the operad composition. The cells are indexed by trees with edges of two colors and vertices labelled by cells of the cacti operad. This answers ... More
Random Čech Complexes on Manifolds with BoundaryJun 18 2019Let $M$ be a compact, unit volume, Riemannian manifold with boundary. In this paper we study the homology of a random \v{C}ech-complex generated by a homogeneous Poisson process in $M$. Our main results are two asymptotic threshold formulas, an upper ... More
Homology of Hurwitz spaces and the Cohen--Lenstra heuristic for function fields (after Ellenberg, Venkatesh, and Westerland)Jun 18 2019Ellenberg, Venkatesh, and Westerland have established a weak form of the function field analogue of the Cohen--Lenstra heuristic, on the distribution of imaginary number fields with $\ell$-parts of their class groups isomorphic to a fixed group. They ... More
Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino term of the M5-braneJun 18 2019The full 6d Wess-Zumino term in the action functional for the M5-brane is anomalous as traditionally defined. What has been missing is a condition implying the higher analogue of level quantization familiar from the 2d Wess-Zumino term. We prove that ... More
On a generalization of Inoue and Oeljeklaus-Toma manifoldsJun 18 2019In this paper we construct a family of complex analytic manifolds that generalize Inoue surfaces and Oeljeklaus-Toma manifolds. To a matrix $M$ in $SL(N,\mathbb{Z})$ satisfying some mild conditions on its characteristic polynomial we associate a manifold ... More
The Complex of Affinely Commutative SetsJun 17 2019We show that for some classes of groups $G$, the homotopy fiber $E_{\text{com}} G$ of the inclusion of the classifying space for commutativity $B_{\text{com}} G$ into the classifying space $BG$, is contractible if and only if $G$ is abelian. We show this ... More
Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic limitJun 17 2019We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected ... More
On infinite iterations of the functor of idempotent probability measuresJun 17 2019In this paper we establish that the functor of idempotent probability measures acting in the category of compacta and their continous mappings is perfect metrisable
A homotopy Lie formula for the p-adic Dwork Frobenius operatorJun 15 2019We give a modern deformation theoretic interpretation of Dwork's theory of the zeta function of a smooth projective complete intersection variety $X$ over a finite field. Using this interpretation, we explicitly construct a dgla (differential graded Lie ... More
Galois descent criteriaJun 14 2019This paper gives an introduction to homotopy descent, and its applications in algebraic $K$-theory computations for fields. On the \'etale site of a field, a fibrant model of a simplicial presheaf can be constructed from naive Galois cohomological objects ... More
Cosimplicial spaces and cocyclesJun 14 2019Standard results from non-abelian cohomology theory specialize to a theory of torsors and stacks for cosimplicial groupoids. The space of global sections of the stack completion of a cosimplicial groupoid $G$ is weakly equivalent to the Bousfield-Kan ... More
Combinatorial homotopy theory for operadsJun 14 2019We introduce an explicit combinatorial characterization of the minimal model ${\cal O}_{\infty}$ of the coloured operad ${\cal O}$ encoding non-symmetric operads. In our description of ${\cal O}_{\infty}$, the spaces of operations are defined in terms ... More
Topological Data Analysis with $ε$-net Induced Lazy Witness ComplexJun 14 2019Topological data analysis computes and analyses topological features of the point clouds by constructing and studying a simplicial representation of the underlying topological structure. The enthusiasm that followed the initial successes of topological ... More
Topological Data Analysis with $ε$-net Induced Lazy Witness ComplexJun 14 2019Jun 19 2019Topological data analysis computes and analyses topological features of the point clouds by constructing and studying a simplicial representation of the underlying topological structure. The enthusiasm that followed the initial successes of topological ... More
The homotopy invariance of dihedral homology of involutive $A_\infty$-algebras over ringsJun 14 2019The dihedral homology functor $HD:A_\infty^{{\rm inv}}(K)\to GrM(K)$ from the category $A_\infty^{{\rm inv}}(K)$ of involutive $A_\infty$-algebras over any commutative unital ring $K$ to the category $GrM(K)$ of graded $K$-modules is constructed. Further, ... More
A Fundamental Group for Digital ImagesJun 14 2019We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously ... More
Enough vector bundles on orbispacesJun 13 2019We show that every coarsely finite-dimensional orbispace with isotropy groups of bounded order has enough (finite-dimensional) vector bundles. It follows that the K-theory of finite-dimensional vector bundles on compact orbispaces is well behaved. Global ... More
An extensional $λ$-model with $\infty$-grupoid structureJun 13 2019From a topological space, a set with $\infty$-grupoid structure is built and this construction is applied to the case of ordered sets equipped with the Scott topology. The main purpose is to project the $\lambda$-model $D_\infty$ of Dana Scott to an extensional ... More
Simplicial equations for the moduli space of stable rational curvesJun 12 2019In this, largely expository, note, we show how the simplicial structure of the moduli spaces of stable rational curves with marked points allows to produce explicit equations for these spaces. The key argument is an elementary combinatorial statement ... More
Homology, lower central series, and hyperplane arrangementsJun 12 2019We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. ... More
Homological Connectivity in Random Čech ComplexesJun 11 2019Jun 13 2019We study the homology of random \v{C}ech complexes generated by a homogeneous Poisson process. We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology "stabilizes" and becomes isomorphic to that ... More
Homological Connectivity in Čech ComplexesJun 11 2019We study the homology of random \v{C}ech complexes generated by a homogeneous Poisson process. We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology "stabilizes" and becomes isomorphic to that ... More