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Classifying spaces for commutativity of low-dimensional Lie groupsFeb 10 2018For each of the groups $G = O(2), SU(2), U(2)$, we compute the integral and $\mathbb{F}_2$-cohomology rings of $B_\text{com} G$ (the classifying space for commutativity of $G$), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy ... More
An explicit symmetric DGLA model of a triangleFeb 08 2018We give explicit formulae for a differential graded Lie algebra (DGLA) model of the triangle which is symmetric under the geometric symmetries of the cell. This follows the work of Lawrence-Sullivan on the (unique) DGLA model of the interval and of Gadish-Griniasty-Lawrence ... More
Boardman's whole-plane obstruction group for Cartan-Eilenberg systemsFeb 08 2018Each extended Cartan--Eilenberg system $(H, \partial)$ gives rise to two exact couples and one spectral sequence. We show that the canonical colim-lim interchange morphism associated to $H$ is a surjection, and that its kernel is isomorphic to Boardman's ... More
On characteristic classes of exotic manifold bundlesFeb 07 2018Given a closed simply-connected manifold $M$ of dimension $2n\neq4$, we compare the ring of characteristic classes of smooth oriented bundles with fibre $M$ to the respective ring resulting from replacing $M$ by the connected sum $M\sharp\Sigma$ with ... More
Maurer-Cartan moduli and theorems of Riemann-Hilbert typeFeb 07 2018We study Maurer-Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We prove, in several ... More
Infinite series in cohomology: attractors and Conley indexFeb 07 2018In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $f \colon U \subseteq \mathbb{R}^d \to \mathbb{R}^d$ by analyzing the topological structure of their unstable manifold. We provide a simple ... More
The equivariant cohomology ring of a cohomogeneity-one actionFeb 07 2018We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group.
Tautological classes and smooth bundles over BSU(2)Feb 06 2018For a Lie group G and a smooth manifold W, we study the difference between smooth actions of G on W and bundles over the classifying space of G with fiber W and structure group Diff(W). In particular, we exhibit smooth manifold bundles over BSU(2) that ... More
Cellular Cohomology in Homotopy Type TheoryFeb 06 2018We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many ... More
Quasitoric stably normally split manifoldsFeb 06 2018This paper is devoted to the construction of manifolds, s.t. any complex vector bundle over it is stably totally split. A quasitoric manifold shares this property iff the respective stable normal vector bundle is totally split. Stably complex manifolds ... More
Crossed extensions and equivalences of topological 2-groupoidsFeb 06 2018We provide concrete models for generalized morphisms and Morita equivalences of topological 2-groupoids by introducing the notions of crossings and crossed extensions of groupoid crossed modules. A systematic study of these objects is elaborated and an ... More
Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrumFeb 06 2018We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results ... More
Topological symmetries of simply-connected four-manifolds and actions of automorphism groups of free groupsFeb 06 2018Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ ... More
Listening to the cohomology of graphsFeb 05 2018We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G which is in ... More
The role of Coulomb branches in 2D gauge theoryJan 30 2018I give a simple construction of certain Coulomb branches $C_{3,4}(G;E)$ of gauge theory in 3 and 4 dimensions defined by Nakajima et al. for a compact Lie group $G$ and a polarisable quaternionic representation $E$. The manifolds $C(G; 0)$ are abelian ... More
Couniversal spaces which are equivariant commutative ring spectraJan 29 2018The note identifies which which couniversal spaces have suspension spectra equivalent to commutative orthogonal ring G-spectra for a compact Lie group G. These are precisely those whose cofamily is closed under passage to finite index subgroups. Equivalently ... More
Some results on extension of maps and applicationsJan 29 2018This paper concerns extension of maps using obstruction theory under a non classical viewpoint. It is given a classification of homotopy classes of maps and as an application it is presented a simple proof of a theorem by Adachi about equivalence of vector ... More
Affine Grassmannians in A^1-algebraic topologyJan 25 2018Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If k is infinite and G is split reductive, ... More
Persistent Betti numbers of random Čech complexesJan 25 2018We study the persistent homology of random \v{C}ech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we ... More
Computations of Orbits for the Lubin-Tate RingJan 23 2018In this note, we take a direct approach to computing the orbits for the action of the automorphism group $\mathbb{G}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that that $(R_2/p)_{\mathbb{G}_2} \cong ... More
A Guide for Computing Stable Homotopy GroupsJan 23 2018This paper contains an overview of background from stable homotopy theory used by Freed-Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and to ... More
Incidence bicomodules, Möbius inversion, and a Rota formula for infinity adjunctionsJan 23 2018In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they are certain ... More
Efficient algorithms for computing a minimal homology basisJan 21 2018Efficient computation of shortest cycles which form a homology basis under $\mathbb{Z}_2$-additions in a given simplicial complex $\mathcal{K}$ has been researched actively in recent years. When the complex $\mathcal{K}$ is a weighted graph with $n$ vertices ... More
The motivic Mahowald invariantJan 18 2018Feb 06 2018The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. We study the Mahowald invariant in the setting of motivic stable homotopy theory over $Spec(\mathbb{C})$. ... More
Freudenthal theorem and spherical classes in $H_*QS^0$Jan 18 2018Jan 22 2018This note is on spherical classes in $H_*(QS^0;k)$ when $k=\mathbb{Z},\mathbb{Z}/p$ with a special focus on the case of $p=2$ related to Curtis conjecture. We apply Freudenthal theorem to prove a vanishing result for the Hurewicz image of elements in ... More
Local Coordinate Spaces: a proposed unification of manifolds and fiber bundles, and associated machineryJan 17 2018This paper presents a unified view of manifolds and fiber bundles, which, while superficially different, have strong parallels. It introduces the notions of an m-atlas and of a local coordinate space, and shows that special cases are equivalent to fiber ... More
A Simplicial Approach to Stratified Homotopy TheoryJan 15 2018In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial combinatorial model category. ... More
2D problems in groupsJan 13 2018Jan 22 2018We investigate a conjecture about stabilisation of deficiency in finite index subgroups and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem. We verify the pro-$p$ version of the conjecture, as well as its higher dimensional abstract ... More
Integral cohomology of configuration spaces of the sphereJan 12 2018We compute the cohomology of the unordered configuration spaces of the sphere $S^2$ with integral and with $\mathbb{Z}/p \mathbb{Z}$-coefficients using a cell complex by Fuks, Vainshtein and Napolitano.
Stability in the homology of Deligne-Mumford compactificationsJan 11 2018Using the the theory of FS^op modules, we study the asymptotic behavior of the homology of $\overline M_{g,n}$, the Deligne--Mumford compactification of the moduli space of curves, for $n >> 0$. An FS^op module is a contravariant functor from the category ... More
Smooth Version of Johnson's Problem Concerning Derivations of Group AlgebrasJan 10 2018A description of the algebra of outer derivations of a group algebra of a finitely presented discrete group is given in terms of the Cayley complex of the groupoid of the adjoint action of the group. This task is a smooth version of Johnson's problem ... More
A-infinity structures and Massey productsJan 09 2018We show how to detect and recover higher Massey products on the cohomology $H$ of a differential graded algebra using $A_\infty$ structures induced on $H$ via homotopy transfer techniques.
A combinatorial model for the free loop fibrationDec 06 2017We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration $\Omega Y\rightarrow \Lambda Y\rightarrow Y$ over the geometric realization $Y=|X|$ of a path connected simplicial ... More
Grothendieck's homotopy theory, polynomial monads and delooping of spaces of long knotsDec 04 2017We extend some classical results - such as Quillen's Theorem A, the Grothendieck construction, and the characterisation of homotopically cofinal functors - from the homotopy theory of small categories to polynomial monads and their algebras. As an application ... More
Non-unital polygraphs form a presheaf categoriesNov 02 2017We prove, as claimed by A.Carboni and P.T.Johnstone, that the category of non-unital polygraphs, i.e. polygraphs where the source and target of each generator are not identity arrows, is a presheaf category. More generally we develop a new criterion for ... More
Homological stability of topological moduli spacesOct 23 2017Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded ... More
On quasi-isometric nilpotent Lie groupsOct 12 2017Nov 20 2017In this article we provide evidence for a well-known conjecture which states that quasi-isometric simply-connected nilpotent Lie groups are isomorphic. We do so by constructing new examples which are rigid in the sense that whenever they are quasi-isometric ... More
Equivariant complex bundles, fixed points and equivariant unitary bordismOct 02 2017We study the fixed points of the universal $G$-equivariant $n$-dimensional complex vector bundle and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller dimension. We use this decomposition to describe ... More
A remark on the group-completion theoremSep 07 2017Suppose that $M$ is a topological monoid satisfying $\pi_0M=\mathbb{N}$ to which the McDuff-Segal group-completion theorem applies. This implies that a certain map $f: \mathbb{M}_{\infty}\rightarrow \Omega BM$ defined on an infinite mapping telescope ... More
Twisted equivariant K-theory of compact Lie group actions with maximal rank isotropySep 04 2017We consider twisted equivariant K--theory for actions of a compact Lie group $G$ on a space $X$ where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequence \`a la Segal has a simple $E_2$--term ... More
When is the heart of a t-structure a Grothendieck category?Aug 24 2017Let $\mathcal D$ be a triangulated category endowed with a $t$-structure $\mathfrak t=(\mathcal U,\Sigma \mathcal V)$ and denote by $\mathcal H:=\mathcal U\cap \Sigma\mathcal V$ its heart. In this paper we study the following well-known problem: Under ... More
The $K(π, 1)$-problem for restrictions of complex reflection arrangementsAug 17 2017In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all Coxeter groups. This follows from Deligne's seminal work ... More
Derivations of Group AlgebrasAug 16 2017In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.
Symplectic spinors and Hodge theoryAug 07 2017Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the ... More
Hyperplane Equipartitions Plus ConstraintsAug 01 2017Oct 09 2017While equivariant methods have seen many fruitful applications in geometric combinatorics, their inability to answer the now settled Topological Tverberg Conjecture has made apparent the need to move beyond the use of Borsuk-Ulam type theorems alone. ... More
Bialgebraic proof of the existence of cup-product in the cohomology of racks and quandlesJul 04 2017We retrieve the graded commutative algebra structure of rack and quandle cohomology by purely algebraic means.
Metric reconstruction via optimal transportJun 15 2017Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ ... More
Computational Tools for Topological CoHochschild HomologyJun 06 2017In recent work, Hess and Shipley defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in ... More
A combinatorial model for the path fibrationJun 03 2017We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set $X$ ... More
Model topoi and motivic homotopy theoryApr 27 2017Given a small simplicial category $\mathcal{C}$ whose underlying ordinary category is equipped with a Grothendieck topology $\tau$, we construct a model structure on the category of simplicially enriched presheaves on $\mathcal{C}$ where the weak equivalences ... More
The $L^2$-torsion polytope of amenable groupsApr 24 2017Jan 29 2018We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among $G$-CW-complexes ... More
Singular Hochschild cohomology and algebraic string operationsMar 11 2017Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns ... More
Dijkgraaf-Witten $Z_2$-invariants for Seifert manifoldsFeb 25 2017In this short paper we compute the values of Dijkgraaf-Witten invariants over $Z_2$ for all orientable Seifert manifolds with orientable bases.
Motivic zeta functions and infinite cyclic coversFeb 21 2017We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) a rational function in $K_0({\rm Var}^{\hat \mu}_{\mathbb{C}})[\mathbb{L}^{-1}]$, ... More
On affine Tverberg-type results without continuous generalizationFeb 17 2017Recent progress building on the groundbreaking work of Mabillard and Wagner has shown that there are important differences between the affine and continuous theory for Tverberg-type results. These results aim to describe the intersection pattern of convex ... More
Cubicial rigidification, the cobar construction, and the based loop spaceDec 14 2016Dec 15 2016We prove the following generalization of a classical result of Adams: for any pointed and connected topological space $(X,b)$, that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular ... More
Cohomology of Polychromatic Configuration Spaces of Euclidean SpaceDec 08 2016Recently, the homology and cohomology of non-k-overlapping discs, or, equivalently, no k-equal subspaces of Euclidean space, were calculated by Dobrinskaya and Turchin. We calculate the homology and cohomology of two classes of more general spaces: decreasing ... More
The topological complexity of the Klein bottle equals 5Dec 08 2016We use obstruction theory to prove that the topological complexity of the Klein bottle equals 5, a new result.
A-infinity resolutions and the Golod property for monomial ringsDec 08 2016Let R=S/I be a monomial ring whose minimal free resolution F is rooted. We describe an A-infinity algebra structure on F. Using this structure, we show that R is Golod if and only if the product on Tor^S(R,k) vanishes. Furthermore, we give a necessary ... More
The $v_n$-periodic Goodwillie tower on Wedges and CofibresDec 08 2016We introduce general methods to analyse the Goodwillie tower of the identity functor on a wedge $X \vee Y$ of spaces (using the Hilton-Milnor theorem) and on the cofibre $\mathrm{cof}(f)$ of a map $f: X \rightarrow Y$. We deduce some consequences for ... More
The abstract cotangent complex and Quillen cohomology of enriched categoriesDec 08 2016In his fundamental work, Quillen developed the theory of the cotangent complex as a universal abelian derived invariant, and used it to define and study a canonical form of cohomology, encompassing many known cohomology theories. Additional cohomology ... More
Tangent categories of algebras over operadsDec 08 2016The abstract cotangent complex formalism, as developed in the $\infty$-categorical setting by Lurie, brings together classical notions such as parametrized spectra, obstruction theory and deformation theory in a unified setting. When the $\infty$-category ... More
Faithful Actions from Hyperplane ArrangementsDec 08 2016An axiomatic framework is developed, under which the tilting modules of an algebra produce a faithful group action on its derived category. As a consequence, if X is a quasi-projective 3-fold admitting a flopping contraction, then the fundamental group ... More
Homotopy theory of unital algebrasDec 07 2016This paper provides an extensive study of the homotopy theory of types of algebras with units, like unital associative algebras or unital commutative algebras for instance. To this purpose, we endow the Koszul dual category of curved coalgebras, where ... More
Computations in $C_{pq}$-Bredon cohomologyDec 07 2016In this paper, we compute the $RO(C_{pq})$-graded cohomology of $C_{pq}$-orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought ... More
Local and global coincidence homology classesDec 07 2016For two differentiable maps between two manifolds of possibly different dimensions, the local and global coincidence homology classes are introduced and studied by Bisi- Bracci-Izawa-Suwa (2016) in the framework of Cech-de Rham cohomology. We take up ... More
The unramified inverse Galois problem and cohomology rings of totally imaginary number fieldsDec 06 2016We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply this obstruction theory to give an infinite family of ... More
Rational torus-equivariant stable homotopy IV: thick tensor ideals and the Balmer spectrum for finite spectraDec 06 2016We classify thick tensor ideals of finite objects in the category of rational torus-equivariant spectra, showing that they are completely determined by geometric isotropy. This is essentially equivalent to showing that the Balmer spectrum is the set of ... More
A review of geometric, topological and graph theory apparatuses for the modeling and analysis of biomolecular dataDec 06 2016Geometric, topological and graph theory modeling and analysis of biomolecules are of essential importance in the conceptualization of molecular structure, function, dynamics, and transport. On the one hand, geometric modeling provides molecular surface ... More
Kimura-finiteness of quadric fibrations over smooth curvesDec 05 2016In this short note, making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite.
Perverse sheaves and the reductive Borel-Serre compactificationDec 04 2016We briefly introduce the theory of perverse sheaves with special attention to the topological situation where strata can have odd dimension. This is part of a project to use perverse sheaves on the topological reductive Borel-Serre compactification of ... More
Quasi-Elliptic Cohomology and its Power OperationsDec 03 2016Quasi-elliptic cohomology is a slightly different variant of Tate K-theory. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. This theory has power operations. ... More
On topological Hochschild homology of the $K(1)$-local sphereDec 02 2016We compute topological Hochschild homology mod $p$ and $v_1$ of the connective cover of the $K(1)$ local sphere spectrum for all primes $p\ge 3$. This is accomplished using a May-type spectral sequence in topological Hochschild homology constructed from ... More
A May-type spectral sequence for higher topological Hochschild homologyDec 02 2016Given a filtration of a commutative monoid $A$ in a symmetric monoidal stable model category $\mathcal{C}$, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the ... More
On the covering type of a spaceDec 02 2016We introduce the notion of the "covering type" of a space, which is more subtle that the notion of Lusternik Schnirelman category. It measures the complexity of a space which arises from coverings by contractible subspaces whose non-empty intersections ... More
On the Cohomology of the Classifying Spaces of Projective Unitary GroupsDec 01 2016In this paper we construct a spectral sequence converging to the integral cohomology ring $H^{*}(\mathbf{B}PU_{n}; \mathbb{Z})$ for any $n>1$, where $\mathbf{B}PU_{n}$ is the classifying space of the projective unitary group of order $n$. We use this ... More
Maximal Sections of Sheaves of Data over an Abstract Simplicial ComplexDec 01 2016We employ techniques from topological data analysis to model sensor networks. Our approach to sensor integration uses the topological method of sheaves over cell complexes. The internal consistency of data from individual sensors is determined by a set ... More
On the moduli space of flat symplectic surface bundlesNov 30 2016In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology ... More
Symmetric products of a real curve and the moduli space of Higgs bundlesNov 29 2016Consider a Riemann surface $X$ of genus $g \geq 2$ equipped with an antiholomorphic involution $\tau$. This induces a natural involution on the moduli space $M(r,d)$ of semistable Higgs bundles of rank $r$ and degree $d$. If $D$ is a divisor such that ... More
Symmetric products of a real curve and the moduli space of Higgs bundlesNov 29 2016Nov 30 2016Consider a Riemann surface $X$ of genus $g \geq 2$ equipped with an antiholomorphic involution $\tau$. This induces a natural involution on the moduli space $M(r,d)$ of semistable Higgs bundles of rank $r$ and degree $d$. If $D$ is a divisor such that ... More
Examples of non-algebraic classes in the Brown-Peterson towerNov 28 2016For every $n\ge 0$, we construct classes in the Brown-Peterson cohomology $BP\langle n \rangle$ of smooth projective complex algebraic varieties which are not in the image of the cycle map from the corresponding motivic Brown-Peterson cohomology. This ... More
The mod 2 homology of free spectral Lie algebrasNov 27 2016The Goodwillie derivatives of the identity functor on pointed spaces form an operad in spectra. Adapting a definition of Behrens, we introduce mod 2 homology operations for algebras over this operad and prove these operations account for all the mod 2 ... More
Open-closed modular operads, Cardy condition and string field theoryNov 26 2016We prove that the modular operad of diffeomorphism classes of Riemann surfaces with both `open' and `closed' boundary components, in the sense of string field theory, is the modular completion of its genus 0 part quotiented by the Cardy condition. We ... More
An algebraic model for rational SO(3)-spectraNov 25 2016Greenlees established an equivalence of categories between the homotopy category of rational SO(3)-spectra and the derived category DA(SO(3)) of a certain abelian category. In this paper we lift this equivalence of homotopy categories to the level of ... More
The second cohomology of nilpotent orbits in classical Lie algebrasNov 25 2016The second de Rham cohomology of real nilpotent orbits in non-compact classical simple real Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology of nilpotent orbits in non-compact classical simple real Lie algebras are computed; ... More
A geometric proof of Lück's vanishing theorem for the first $L^2$-Betti number of the total space of a fibrationNov 24 2016A significant theorem of L\"uck says that the first $L^2$-Betti number of the total space of a fibration vanishes under some conditions on the fundamental groups. The proof is based on constructions on chain complexes. In the present paper, we translate ... More
Integral String Lie Algebra Structure of SpheresNov 23 2016Chas and Sullivan introduced string homology, which is the equivariant homology of the loop space with the $S^1$ action on loops by rotation. Craig Westerland computed the string homology for spheres with coefficients in $\mathbb{Z} /2\mathbb{Z}$ and ... More
Integral String Lie Algebra Structure of SpheresNov 23 2016Dec 08 2016Chas and Sullivan introduced string homology, which is the equivariant homology of the loop space with the $S^1$ action on loops by rotation. Craig Westerland computed the string homology for spheres with coefficients in $\mathbb{Z} /2\mathbb{Z}$ and ... More
On Stiefel-Whitney classes of vector bundles over real Stiefel ManifoldsNov 23 2016In this article we show that there are at most two integers up to $2(n-k)$, which can occur as the degrees of nonzero Stiefel-Whitney classes of vector bundles over the Stiefel manifold $V_k(\mathbb{R}^n)$. In the case when $n> k(k+4)/4$, we show that ... More
Pullback Crossed Modules in the Category of RacksNov 21 2016In this paper, we define the pullback crossed modules in the category of racks which mainly based on a pullback diagram of rack morphisms with extra crossed module data on some of its arrows. Furthermore we prove that the conjugation functor, which is ... More
Power operations for $\text{H}\underline{\mathbb{F}}_2$ and a cellular construction of $\text{BP}\mathbf{R}$Nov 21 2016We study some power operations for ordinary $C_2$-equivariant homology with coefficients in the constant Mackey functor $\underline{\mathbb{F}}_2$. In addition to a few foundational results, we calculate the action of these power operations on a $C_2$-equivariant ... More
A lemma for microlocal sheaf theory in the $\infty$-categorical settingNov 21 2016Microlocal sheaf theory of \cite{KS90} makes an essential use of an extension lemma for sheaves due to Kashiwara, and this lemma is based on a criterion of the same author giving conditions in order that a functor defined in $\mathbb{R}$ with values in ... More
T-Duality from super Lie n-algebra cocycles for super p-branesNov 20 2016We compute the $L_\infty$-theoretic dimensional reduction of the F1/D$p$-brane super $L_\infty$-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting ... More
Twisted Higher Smooth TorsionNov 18 2016In this paper we extend Badzioch's, Dorabiala's, and Williams' definition of cohomological higher smooth torsion to a twisted cohomological higher torsion invariant. Additionally, we show that this still satisfies geometric additivity and transfer, and ... More
Maps of simplicial spectra whose realizations are cofibrationsNov 18 2016Given a map of simplicial topological spaces, mild conditions on degeneracies and the levelwise maps imply that the geometric realization of the simplicial map is a cofibration. These conditions are not formal consequences of model category theory, but ... More
Nilpotent $n$-tuples in $SU(2)$Nov 18 2016Let $F_n/\Gamma^q_n$ denote the finitely generated free $q$-nilpotent group. We describe the connected components of the spaces of homomorphisms $Hom(F_n/\Gamma^q_n,SU(2))$, $Hom(F_n/\Gamma^q_n,SO(3))$ and $Hom(F_n/\Gamma^q_n,U(2))$. We also describe ... More
Recent developments on noncommutative motivesNov 16 2016Written for the proceedings of the second Mid-Atlantic Topology Conference, held at Johns Hopkins University, this survey covers some of the recent developments on noncommutative motives and their applications. Among other topics, we compute the additive ... More
Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knotsNov 15 2016We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q:=H_q(N;Z)$. Our main result is a complete readily calculable classification of embeddings $N\to R^7$, up to the equivalence relation ... More
Computations of the Structure of the Goldman Lie Algebra for the TorusNov 15 2016We consider the structure of the Goldman Lie algebra for the closed torus, and show that it is finitely generated over the rationals. We also consider other traditional Lie algebra structures and determine that the Goldman Lie algebra for the torus is ... More
The universal $n$-pointed surface bundle only has $n$ sectionsNov 14 2016The classifying space BDiff$(S_{g,n})$ of the orientation-preserving diffeomorphism group of the surface $S_{g,n}$ of genus $g>1$ with $n$ ordered marked points has a universal bundle \[ S_g \to \text{UDiff}(S_{g,n})\xrightarrow{\pi}\text{BDiff}(S_{g,n}). ... More