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Complexity reduction for path categoriesSep 18 2019This paper displays complexity reduction techniques for calculations of path categories (or fundamental categories) P(K) for finite simplicial and cubical complexes K. The central technique involves identifying inclusions of complexes for which the induced ... More

Pro-equivalences of diagramsSep 18 2019This paper presents a model structure for natural transformations of diagrams of simplicial presheaves of a fixed shape, in which the weak equivalences are defined by analogy with pro-equivalences between pro-objects.

Path categories and quasi-categoriesSep 18 2019This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids effectively replace ... More

Homotopy Types of Abstract Elementary ClassesSep 17 2019We prove that for any homotopy type $X$, there is an elementary class $\calC$, with joint embedding, almagamation and no maximal models such that the classifying space realizes the homotopy type $X$. We provide a few explicit examples.

Representation stability for pure braid group Milnor fibersSep 17 2019We prove a representation stability result for the Milnor fiber associated to the pure braid group. Our result connects previous work of Simona Settepenella to representation stability in the sense of Church--Ellenberg--Farb, answering a question of Graham ... More

Effective computation of degree bounded minimal models for GCDA'sSep 17 2019Given a finitely presented Graded Commutative Differential Algebra (GCDA), we present a method to compute its minimal model, together with a map that is a quasi-isomorphism up to a given degree. The method works by adding generators one by one. We also ... More

Persistence B-Spline Grids: Stable Vector Representation of Persistence Diagrams Based on Data FittingSep 17 2019Over the last decades, many attempts have been made to optimally integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram ... More

Elliptic classes, McKay correspondence and theta identitiesSep 16 2019We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to equivariant local situation. We study theta function identities having geometric origin. In the ... More

On vanishing of all fourfold products of the Ray classes in symplectic cobordismSep 16 2019This note provides certain computations with transfer associated with projective bundles of Spin vector bundles. One aspect is to revise the proof of the main result of \cite{B} which says that all fourfold products of the Ray classes are zero in symplectic ... More

Disproportionate divisionSep 16 2019We study the disproportionate version of the classical cake-cutting problem: how efficiently can we divide a cake, here $[0,1]$, among $n$ agents with different demands $\alpha_1, \alpha_2, \dots, \alpha_n$ summing to $1$? When all the agents have equal ... More

Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospansSep 16 2019We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference ... More

The octonionic projective planeSep 16 2019This small note, without claim of originality, constructs the projective plane over the octonionic numbers and recalls how this can be used to rule out the existence of higher-dimensional real division algebras, using Adams' solution of the Hopf invariant ... More

Dense products in fundamental groupoidsSep 15 2019Infinitary 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 ... More

Scattered products in fundamental groupoidsSep 15 2019Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. We prove that the well-definedness of products indexed by a scattered linear order in the fundamental ... More

A Variation of the Goldman-Millson Theorem for Filtered $L_\infty$ AlgebrasSep 14 2019In this paper, we extend the Goldman-Millson Theorem for $L_\infty$ algebras. We consider $L$ and $\tilde{L}$, two $L_\infty$ algebras endowed with descending, bounded above and complete filtrations compatible with the $L_\infty$ structures and $U:L \rightarrow ... More

From Ohkawa to strong generation via approximable triangulated categories -- a variation on the theme of Amnon Neeman's Nagoya lecture seriesSep 14 2019This survey stems from Amnon Neeman's lecture series at Ohakawa's memorial workshop. Starting with Ohakawa's theorem, this survey intends to supply enough motivation, background and technical details to read Neeman's recent papers on his "approximable ... More

An enriched count of the bitangents to a smooth plane quartic curveSep 12 2019Recent work of Kass--Wickelgren gives an enriched count of the $27$ lines on a smooth cubic surface over arbitrary fields. Their approach using $\mathbb{A}^1$-enumerative geometry suggests that other classical enumerative problems should have similar ... More

Refinement invariance of intersection (co)homologiesSep 12 2019We study the refinement invariance of several intersection (co)homologies existing in the literature. These (co)homologies have been introduced in order to establish the Poincar\'e Duality in variousl contexts. We found the classical topological invariance ... More

Bidirectional Sequential Motion PlanningSep 12 2019We define a simpler notion of symmetric topological complexity more ad hoc to the motion planning problem which was the original motivation for the definition of topological complexity. This is a homotopy invariant that we call bidirectional topological ... More

3d-printing Identification Spaces of the SquareSep 12 2019We describe three identification spaces of the square, interesting choices of immersion into $\mathbb{R}^3$, and a process to construct 3d-printable models of their parametrizations.

Choreography of divisors on algebraic curvesSep 12 2019For a non-singular real algebraic projective curve, topological restrictions on a closed motion of a simple real divisor in its linear equivalence class are found.

The group Aut and Out of the fundamental group of a closed Sol 3-manifoldSep 11 2019Let $E$ be the fundamental group of a closed Sol 3-manifold. We describe the groups $Aut(E)$ and $Out(E)$. We first consider the case where $E$ is the fundamental group of a torus bundle, and then the case where $E$ is the fundamental group of a closed ... More

Schubert structure operators and K_T(G/B)Sep 11 2019We prove a formula for the structure constants of multiplication of equivariant Schubert classes in both equivariant cohomology and equivariant K-theory of Kac-Moody flag manifolds G/B. We introduce new operators whose coefficients compute these (in a ... More

Localization and nilpotent spaces in A^1-homotopy theorySep 11 2019For a subring $R$ of the rational numbers, we study $R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${\mathbb A}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence ... More

The third Milgram-Priddy class liftsSep 11 2019We show that the third cohomology of the finite general linear group $GL_6(\mathbb{F}_2)$ with trivial mod 2 coefficients is non-zero. The necessarily unique non-trivial element restricts to the third Milgram-Priddy class.

Equivariant stable categories for incomplete systems of transfersSep 10 2019In this paper, we construct incomplete versions of the equivariant stable category; i.e., equivariant stabilization of the category of $G$-spaces with respect to incomplete systems of transfers encoded by an $N_\infty$ operad $\mathcal{O}$. These categories ... More

The Homotopy Types of $SU(4)$-Gauge GroupsSep 10 2019Let $\mathcal{G}_k$ be the gauge group of the principal $SU(4)$-bundle over $S^4$ with second Chern class $k$ and let $p$ be a prime. We show that there is a rational or $p$-local homotopy equivalence $\Omega\mathcal{G}_k\simeq\Omega\mathcal{G}_{k'}$ ... More

The Homotopy Types of $SU(4)$-Gauge GroupsSep 10 2019Sep 11 2019Let $\mathcal{G}_k$ be the gauge group of the principal $SU(4)$-bundle over $S^4$ with second Chern class $k$ and let $p$ be a prime. We show that there is a rational or $p$-local homotopy equivalence $\Omega\mathcal{G}_k\simeq\Omega\mathcal{G}_{k'}$ ... More

Equivariant Morse theory on Vietoris-Rips complexes & universal spaces for proper actionsSep 10 2019We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.

Two-Dimensional Extended Homotopy Field TheoriesSep 09 2019We define $2$-dimensional extended homotopy field theories (E-HFTs) with aspherical targets and classify them. When target is a $K(G,1)$-space, oriented E-HFTs taking values in the symmetric monoidal bicategory of algebras, bimodules, and bimodule maps ... More

On the parametrized Tate construction and two theories of real $p$-cyclotomic spectraSep 09 2019We give a new formula for $p$-typical real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for $p$-typical topological cyclic homology to one involving genuine $C_2$-spectra. To accomplish this, we ... More

Crossing-changeable braids from chromatic configuration spacesSep 09 2019Motivated by the work in [15], this paper deals with the theory of the braids from chromatic configuration spaces. This kind of braids possess the property that some strings of each braid may intersect together and can also be untangled, so they are quite ... More

Definable (co)homology, pro-torus rigidity, and (co)homological classificationSep 09 2019We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact metrizable ... More

Definable (co)homology, pro-torus rigidity, and (co)homological classificationSep 09 2019Sep 14 2019We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact metrizable ... More

Probabilistic Convergence and Stability of Random Mapper GraphsSep 08 2019We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. We first give a categorification of the mapper graph and ... More

Adams-Hilton model and the group of self-homotopy equivalences of a simply connected cw-complexSep 08 2019Let $R$ be a principal ideal domain (PID). For a simply connected CW-complex $X$ of dimension $n$, let $Y$ be a space obtained by attaching cells of dimension $q$ to $X$, $q>n$, and let $A(Y)$ denote an Adams-Hilton model of $Y$. If $\mathcal E(A(Y))$ ... More

Covers in the Canonical Grothendieck TopologySep 08 2019We explore the canonical Grothendieck topology in some specific circumstances. First we use a description of the canonical topology to get a variant of Giraud's Theorem. Then we explore the canonical Grothendieck topology on the categories of sets and ... More

A-module extensionsSep 06 2019Explicit extensions representing cocycles $x \in Ext_{A}^{s,t}(F_2,F_2)$ are useful in calculating Steenrod operations $Sq^i : Ext_{A}^{s,t}(F_2,F_2) \longrightarrow Ext_{A}^{s+i,2t}(F_2,F_2)$ by a method devised by the second author. This can be used ... More

Steenrod operations and A-module extensionsSep 06 2019Sep 12 2019Explicit extensions representing cocycles $x \in Ext_{A}^{s,t}(F_2,F_2)$ are useful in calculating Steenrod operations $Sq^i : Ext_{A}^{s,t}(F_2,F_2) \longrightarrow Ext_{A}^{s+i,2t}(F_2,F_2)$ by a method devised by the second author. This can be used ... More

Complex K-theory of mirror pairsSep 06 2019We formulate some conjectures about the K-theory of symplectic manifolds and their Fukaya categories, and prove some of them in very special cases.

On the vanishing of discrete singular cubical homology for graphsSep 06 2019We prove that if G is a graph without 3-cycles and 4-cycles, then the discrete cubical homology of G is trivial in dimension d, for all d\ge 2. We also construct a sequence { G_d } of graphs such that this homology is non-trivial in dimension d for d\ge ... More

On the top dimensional cohomology groups of congruence subgroups of $\text{SL}_n(\mathbb{Z})$Sep 05 2019Let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of $\Gamma_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let ... More

Knots with Prism Manifold SurgeriesSep 05 2019Ballinger et al. have determined the list of all prism manifolds that are possibly realizable by Dehn surgeries on knots in $S^3$. In this paper, we explicitly find braid words of primitive/Seifert-fibered knots on which surface slope surgeries yield ... More

Polyhedral products, and relations in the commutator subgroup of a right-angled Coxeter groupSep 04 2019We give a criterion of the existence of a presentation with a single relation for the commutator subgroup $RC_{\mathscr{K}}'$ of a right-angled Coxeter group $RC_{\mathscr{K}}$. Namely, we prove that $RC_{\mathscr{K}}'$ is a one-relator group if and only ... More

The Witt group of real surfacesSep 04 2019Let $V$ be an algebraic variety defined over $\mathbb R$, and $V_{top}$ the space of its complex points. We compare the algebraic Witt group $W(V)$ of symmetric bilinear forms on vector bundles over $V$, with the topological Witt group $WR(V_{top})$ of ... More

The dualizing module and top-dimensional cohomology group of $\text{GL}_n(\mathcal{O})$Sep 03 2019For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast ... More

Equivariant Cohomological Rigidity of Topological Contact Toric ManifoldsSep 03 2019We introduce the category of topological contact toric manifolds which is a topological generalization of compact connected contact toric manifolds, and study their basic properties. Our main theorem says that two topological contact toric manifolds are ... More

Spaces of knotted circles and exotic smooth structuresSep 03 2019Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension $n$ that are homeomorphic. We prove that the spaces of smooth knots $Emb(S^1, N_1)$ and $Emb(S^1, N_2)$ have the same homotopy $(2n-7)$-type. In the 4-dimensional case this means that ... More

The lemmas of Alexander and SpernerSep 03 2019Alexander's lemma is a version of Sperner's lemma published by Alexander two years earlier than Sperner's paper. The present paper is devoted to a modern but elementary exposition of lemmas of Alexander and Sperner and their main topological applications: ... More

On PM-mapping class monoidsSep 03 2019In this paper, we introduce PM-mapping class monoids. Braid groups and mapping class groups have many features in common. Similarly to the notion of braid PM-monoid, PM-mapping class monoid is defined. This construction is an analogy of inverse mapping ... More

Filtration of cohomology via semi-simplicial spacesSep 01 2019Inspired by Deligne's use of the simplicial theory of hypercoverings in defining mixed Hodge structures, we define the notion of \emph{semi-simplicial filtration} of a family of spaces by some fixed space. A result of the semi-simplicial filtration is ... More

Holomorphic sections of line bundles vanishing along subvarietiesSep 01 2019Let $X$ be a compact normal complex space of dimension $n$, and $L$ be a holomorphic line bundle on $X$. Suppose $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $\tau=(\tau_1,\ldots,\tau_\ell)$ ... More

Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratumAug 30 2019A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. In the published version of "Mapping class group and a global Torelli theorem for hyperkahler manifolds" I made an error based on a wrong quotation ... More

An Orientation Map for Height p-1 Real E TheoryAug 30 2019Let $p$ be an odd prime and let $\mathit{EO} = E_{p-1}^{hC_p}$ be the $C_p$ fixed points of height $p-1$ Morava $E$ theory. We say that a spectrum $X$ has algebraic $\mathit{EO}$ theory if the splitting of $K_*(X)$ as an $K_*[C_p]$-module lifts to a topological ... More

The Lubin-Tate Theory of Configuration Spaces: IAug 29 2019We construct a spectral sequence converging to the Morava $E$-theory of unordered configuration spaces and identify its E$^2$-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the $E$-theory of ... More

Homological epimorphisms and homotopy epimorphismsAug 29 2019We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps ... More

A criterion for the existence of periodic points based on the eigenvalues of maps induced in cohomologyAug 29 2019We present a criterion for the existence of periodic points based on the eigenvalues of maps induced in cohomology for spaces with rational cohomology isomorphic to a tensor product of a graded exterior algebra with generators in odd dimensions and a ... More

Locally type $\text{FP}_n$ and $n$-coherent categoriesAug 28 2019We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type $\text{FP}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type $\text{FP}_n$ ... More

Regular Polygonal Partitions of a Tverberg TypeAug 28 2019A seminal theorem of Tverberg states that any $T(r,d)=(r-1)(d+1) +1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any set of fewer points in $\mathbb{R}^d$ cannot be so divided, ... More

Topological aspects of the dynamical moduli space of rational mapsAug 28 2019We investigate the topology of the space of M\"obius conjugacy classes of degree $d$ rational maps on the Riemann sphere. We show that it is rationally acyclic and we compute its fundamental group. As a byproduct, we also obtain the ranks of some higher ... More

On localized signature and higher rho invariant of fibered manifoldsAug 28 2019Higher index of signature operator is a far reaching generalization of signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator called ... More

Singular chains on Lie groups and the Cartan relations IAug 27 2019Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. We show that the following categories are naturally equivalent. The category $\mathsf{Mod}(C(G))$, of sufficiently smooth modules over the DG-algebra of singular chains on $G$. The ... More

M-Theory anomaly cancellationAug 26 2019We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine generators for the 12-dimensional bordism group of pin manifolds with a w_1-twisted integer lift of w_4; these ... More

Modules over posets: commutative and homological algebraAug 26 2019The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, ... More

External Spanier-Whitehead duality and homology representation theorems for diagram spacesAug 26 2019We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given by taking the homotopy groups of a balanced smash product ... More

Geometric anomaly detection in dataAug 25 2019This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can optimally stratify ... More

Simple homotopy types of independence complexes of graphs involving grid graphsAug 25 2019We show that if a graph $G$ involves a certain square grid graph as a full subgraph, then a certain operation on it yields a simplicial suspension of the independence complex of $G$. This generalizes a result of Csorba. As a corollary, we determine the ... More

The Bracket in the Bar Spectral Sequence for an Iterated Loop SpaceAug 24 2019When $X$ is an associative H-space, the bar spectral sequence computes the homology of the delooping, $H_{*}(BX)$. If $X$ is an $n$-fold loop space for $n\geq2$ this is a spectral sequence of Hopf algebras. Using machinery by Sugawara and Clark, we show ... More

Chromatic Complexity of the Algebraic K-theory of $y(n)$Aug 24 2019The family of Thom spectra $y(n)$ interpolate between the sphere spectrum and the mod two Eilenberg-MacLane spectrum. Computations of Mahowald, Ravenel, and Shick show that the $E_1$-ring spectrum $y(n)$ has chromatic complexity $n$ for $0\le n\le\infty$. ... More

Analyzing Collective Motion with Machine Learning and TopologyAug 24 2019We use topological data analysis and machine learning to study a seminal model of collective motion in biology [D'Orsogna et al., Phys. Rev. Lett. 96 (2006)]. This model describes agents interacting nonlinearly via attractive-repulsive social forces and ... More

First-order homotopical logicAug 23 2019We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this ... More

Six-dimensional gauge theories and (twisted) generalized cohomologyAug 22 2019We consider the global aspects of the 6-dimensional $\mathcal{N}=(1, 0)$ theory arising from the coupling of the vector multiplet to the tensor multiplet. We show that the Yang-Mills field and its dual, when both are abelianized, combine to define a class ... More

C_2-equivariant stable homotopy from real motivic stable homotopyAug 22 2019Sep 11 2019We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant ... More

C_2-equivariant stable homotopy from real motivic stable homotopyAug 22 2019We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant ... More

Good Fibrations through the Modal PrismAug 21 2019Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected points of ... More

Real polynomials with constrained real divisors. I. Fundamental groupsAug 21 2019In the late 80s, V.~Arnold and V.~Vassiliev initiated the study of the topology of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider ... More

Iterated traces in bicategories and Lefschetz theoremsAug 20 2019While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about iterated 2-categorical traces. We show that many Lefschetz-type theorems ... More

Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifoldsAug 20 2019We study the Smith forms of matrices of the form $f(C_g)$ where $f(t),g(t)\in R[t]$, $C_g$ is the companion matrix of the (monic) polynomial $g(t)$, and $R$ is an elementary divisor domain. Prominent examples of such matrices are circulant matrices, skew-circulant ... More

The Topological Complexity of Spaces of Digital Jordan CurvesAug 19 2019This research is motivated by studying image processing algorithms through a topological lens. The images we focus on here are those that have been segmented by digital Jordan curves as a means of image compression. The algorithms of interest are those ... More

Fixed points and semifree bordismAug 19 2019We apply fixed-point techniques to compute the coefficient ring of semifree geometric circle-equivariant complex cobordism with isolated fixed points, recovering a 2004 result of Sinha through 19th-century methods. This should be viewed as an initial ... More

Smooth covers of finite groupsAug 19 2019In the spirit of the homology theory where algebraic and geometric concepts merge, we establish that a natural order preserving condition for covering groups corresponds to having a smooth covering projections between the relevant topological spaces.

Spectral Sequences For Commutative Lie AlgebrasAug 19 2019Aug 20 2019We construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic 2. In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain comparison spectral ... More

Spectral sequences for commutative lie algebrasAug 19 2019We construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic 2. In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain comparison spectral ... More

Directed Homotopy in Non-Positively Curved SpacesAug 19 2019Aug 27 2019A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the particular case of ... More

Directed Homotopy in Non-Positively Curved SpacesAug 19 2019A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the particular case of ... More

Complexes of marked graphs in gauge theoryAug 19 2019We review the gauge graph complexes as defined by Kreimer, Sars and van Suijlekom in "Quantization of gauge fields, graph polynomials and graph homology" and compute their cohomology.

Relative topological surgery exact sequence and additivity of relative higher rho invariantsAug 18 2019In this paper, we define the relative higher $\rho$ invariant for orientation preserving homotopy equivalence between manifolds with boundary in $K$-theory of relative obstruction algebra, i.e relative analytic structure group. We also show that the map ... More

Relative topological surgery exact sequence and additivity of relative higher rho invariantsAug 18 2019Aug 26 2019In this paper, we define the relative higher $\rho$ invariant for orientation preserving homotopy equivalence between manifolds with boundary in $K$-theory of relative obstruction algebra, i.e relative analytic structure group. We also show that the map ... More

Dolbeault cohomology of complex manifolds with torus actionAug 18 2019Sep 16 2019We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds ... More

Dolbeault cohomology of complex manifolds with torus actionAug 18 2019We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM and LVMB-manifolds and, in most generality, complex manifolds ... More

Data and homotopy typesAug 17 2019This paper presents explicit assumptions for the existence of interleaving homotopy equivalences of both Vietoris-Rips and Lesnick complexes associated to an inclusion of data sets. Consequences of these assumptions are investigated on the space level, ... More

Higher Equipments, Double Colimits and Homotopy ColimitsAug 16 2019This document is centered around a main idea: simplicial categories, by which we mean simplicial objects in the category of categories, can be treated as a two-fold categorical structure and their double category theory is homotopically meaningful. The ... More

Brown's Criterion and classifying spaces for familiesAug 15 2019Aug 16 2019Let $G$ be a group and $\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\mathcal{F}$, and for all $H\in ... More

Brown's Criterium and classifying spaces for familiesAug 15 2019Let $G$ be a group and $\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\mathcal{F}$, and for all $H\in ... More

On the homotopy theory of equivariant colored operadsAug 15 2019We build model structures on the category of equivariant simplicial operads with weak equivalences determined by families of subgroups, both in the context of operads with a fixed set of colors and in the context of all colored operads. In particular, ... More

The homotopy theory of coherently commutative monoidal quasi-categoriesAug 14 2019The main objective of this paper is to construct a symmetric monoidal closed model category of coherently commutative monoidal quasi-categories.

The fibre of the degree $3$ map, Anick spaces and the double suspensionAug 14 2019Let $S^{2n+1}\{p\}$ denote the homotopy fibre of the degree $p$ self map of $S^{2n+1}$. For primes $p \ge 5$, work of Selick shows that $S^{2n+1}\{p\}$ admits a nontrivial loop space decomposition if and only if $n=1$ or $p$. Indecomposability in all ... More

Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological ComplexityAug 14 2019We use the homotopy invariance of equivariant principal bundles to prove that the equivariant $\A$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of ... More

Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological ComplexityAug 14 2019Aug 15 2019We use the homotopy invariance of equivariant principal bundles to prove that the equivariant $\A$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of ... More

Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological ComplexityAug 14 2019Aug 20 2019We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann ... More