Mathematics - Algebraic Topology Publications (50)

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Mathematics - Algebraic Topology Publications

We study the categories governing infinity (wheeled) properads. The graphical category, which was already known to be generalized Reedy, is in fact an Eilenberg-Zilber category. A minor alteration the definition of the wheeled graphical category allows us to show that it is a generalized Reedy category. Read More


We give explicit formulae for a DGLA model of the bi-gon 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 its construction uses deeper knowledge of the structure of such models and their localisations for non-simply connected spaces. Read More


We study the action of the orthogonal group on the little $n$-disks operads. As an application we provide small models (over the reals) for the framed little $n$-disks operads. It follows in particular that the framed little $n$-disks operads are formal (over the reals) for $n$ even and coformal for all $n$. Read More


The mapping class group $\Gamma^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of $\mathbb R {\rm P}^2$, we analize the Serre spectral sequence of a fiber bundle $F_k(\mathbb R {\rm P}^2)/\Sigma_k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma^k(\mathbb R {\rm P}^2),1)$ and $F_k(\mathbb R {\rm P}^2)/\Sigma_k$ denotes the configuration space of unordered $k$-tuples of distinct points in $\mathbb R {\rm P}^2$. Read More


Inspired by the analogous result in the algebraic setting (Theorem 1) we show (Theorem 2) that the product $M \times \mathbb{R}P^n$ of a closed and orientable topological manifold $M$ with the $n$-dimensional real projective space cannot be topologically locally flat embedded into $\mathbb{R}P^{m + n + 1}$ for all even $n > m$. Read More


We say that a complete nonsingular toric variety (called a toric manifold in this paper) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds (or Bott towers) are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $\mathrm{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. Read More


We study the homotopy type of the harmonic compactification of the moduli space of a 2-cobordism S with one outgoing boundary component, or equivalently of the space of Sullivan diagrams of type S on one circle. Our results are of two types: vanishing and non-vanishing. In our vanishing results we are able to show that the connectivity of the harmonic compactification increases with the number of incoming boundary components. Read More


We determine the Gross-Hopkins duals of certain higher real K-theory spectra. More specifically, let p be an odd prime, and consider the Morava E-theory spectrum of height n=p-1. It is known, in the expert circles, that for certain finite subgroups G of the Morava stabilizer group, the homotopy fixed point spectra E_n^{hG} are Gross-Hopkins self-dual up to a shift. Read More


We study the relationship between the higher Massey products on the cohomology $H$ of a differential graded algebra, and the $A_\infty$ structures induced on $H$ via homotopy transfer techniques. Read More


Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. Read More


T. Kobayashi [36th Geometry Symposium (1989)] conjectured that a homogeneous space G/H of reductive type does not admit a compact Clifford-Klein form if rank G - rank K < rank H - rank K_H. We solve this conjecture affirmatively. Read More


Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold. Read More


A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type $G_{31}$. Read More


We describe a homotopy-theoretic approach to the moduli of $\Pi$-algebras of Blanc-Dwyer-Goerss using the $\infty$-category $P_{\Sigma}(Sph)$ of product-preserving presheaves on finite-wedges of positive-dimensional spheres, reproving all of their results in this new setting. Read More


In this paper we give a method to construct moment-angle manifolds whose cohomologies are not torsion free. We also give method to describe the corresponding simplicial sphere by its non-faces. Read More


In this paper, we use simplicial Waldhausen theory to show the geometric realization of the topologized category of bounded chain complexes over $\mathbb{F}=\mathbb{C}$ (resp. $\mathbb{R}$) is an infinite loop space that represents connective complex (resp. real) topological $K$-theory. Read More


To every homotopy n-nilpotent group, defined in earlier work by Dwyer and the author, we associate an endofunctor of pointed spaces and prove that it is looped and n-excisive. As a tool we prove that $\Omega P_n({\rm id})$ commutes with sifted colimits of connected spaces. Read More


In this note we present an $\infty$-categorical framework for descent along adjunctions and a general formula for counting conjugates up to equivalence which unifies several known formulae from different fields. Read More


Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, S^n x S^{2n} is a space of type (0, 1) and the one-point union S^n V S^{2n} V S^{3n} is a space of type (0, 0)). It is known that a finite group G which contains Zp + Zp + Zp, p a prime, can not act freely on S^n x S^{2n}. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G can not contain Zp + Zp, p an odd prime. Read More


Recently, P. Pragacz described the ordinary Hall-Littlewood $P$-polynomials by means of push-forwards (Gysin maps) from flag bundles in the ordinary cohomology theory. Together with L. Read More


We establish a combinatorial model for the Boardman--Vogt tensor product of several absolutely free operads, that is free symmetric operads that are also free as $\mathbb{S}$-modules. Our results imply that such a tensor product is always a free $\mathbb{S}$-module, in contrast with the results of Kock and Bremner--Madariaga on hidden commutativity for the Boardman--Vogt tensor square of the operad of non-unital associative algebras. Read More


In this paper, we focus on some models in rational homotopy theory, Sullivan model, Quillen model, C_\infty model, and L_\infty model. We give some connections between them. As an application, we prove the Torus Rank Conjecture. Read More


This paper is a continuation of a previous paper joint with Dennis Sullivan (arXiv:1704.04308). Working in the context of commutative differential graded algebras, we study the ideal of the cohomology classes which can be annihilated by fibrations whose fiber has finite homological dimension. Read More


In this paper, we study the mod(p) motivic cohomology of twisted complete flag varieties over some restricted fields k. Here we take k such that the mod(p) Milnor K-theory KM_i(k)/p=0 for i>3. Read More


We develop a Gabriel-Morita theory for strong monads on pointed monoidal model categories. Assuming that the model category is excisive, i.e. Read More


We prove that the $v_1$-local $G$-equivariant stable homotopy category for $G$ a finite group has a unique $G$-equivariant model at $p=2$. This means that at the prime $2$ the homotopy theory of $G$-spectra up to fixed point equivalences on $K$-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for $K$-local spectra of the second author with the equivariant rigidity result for $G$-spectra of the first author. Read More


This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The topics considered include: fibrancy conditions on topological categories; the effect on classifying spaces of freely adjoining units; approximate notions of units; Quillen's Theorems A and B for non-unital topological categories; the effect on classifying spaces of changing the topology on the space of objects; the Group-Completion Theorem. Read More


We discuss a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. We give several equivalent descriptions, and prove some algebraic and combinatorial properties. In addition, we characterize shuffles in terms of open sets in a topological space associated to a pair of trees. Read More


B\'ar\'any, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex to d-dimensional Euclidean space, if the matroid has sufficiently many disjoint bases. They make a conjecture on the connectivity of k-fold deleted joins of a matroid with many disjoint bases, which would yield a much tighter result - but we provide a counterexample already for the case of k=2, where a tight Tverberg-type theorem would be a topological Radon theorem for matroids. Nevertheless, we prove the topological Radon theorem for the counterexample family of matroids by an index calculation, despite the failure of the connectivity-based approach. Read More


We show that every indexing system arises as the admissible sets of a large, but explicit, categorical $N_\infty$ operad. This positively resolves a conjecture of Blumberg and Hill on the classification of $N_\infty$ operads. Read More


Let $k$ be an algebraically closed field of exponential characteristic $p$. Given any prime $\ell\neq p$, we construct a stable \'etale realization functor $$\text{Spt}(k)\rightarrow \text{Pro}(\text{Spt})^{H\mathbb{Z}/\ell}$$ from the stable $\infty$-category of motivic $\mathbb{P}^1$-spectra over $k$ to the stable $\infty$-category of $(H\mathbb{Z}/\ell)^*$-local pro-spectra (see section 3 for definition). This is induced by the \'etale topological realization functor \'a la Friedlander. Read More


In \cite{Rav84} and \cite{Rav86}, Ravenel introduced sequences of Thom spectra $X(n)$ and $T(n)$ that played an important role in the proof of the nilpotence theorem of Devinatz-Hopkins-Smith \cite{DHS88}. Let $X$ be any one of the Thom spectra $X(n+1)$ or $T(n)$ where $0 \leq n \leq \infty$. We apply the techniques of Lun{\o}e-Nielsen-Rognes to show that the map from the $C_{p^k}$-fixed points of $THH(X)$ to the $C_{p^k}$-homotopy fixed points of $THH(X)$ is a $p$-adic equivalence for all $k \geq 1$. Read More


This series of papers is dedicated to the study of motivic homotopy theory in the context of brave new or spectral algebraic geometry. In Part II we prove a comparison result with the classical motivic homotopy theory of Morel-Voevodsky. This comparison says roughly that any $\mathbf{A}^1$-homotopy invariant cohomology theory in spectral algebraic geometry is determined by its restriction to classical algebraic geometry. Read More


We consider C-homotopy classes of maps from manifolds to spaces and C-homotopy invariants of covers on spaces. We show that the C-homotopy invariants of covers on manifolds is equivalent to the C-homotopy classes of their associated maps. Moreover some C--homotopy groups of spheres and manifolds are determined with applications to homotopy theory of covers on spaces. Read More


We use nonabelian Poincar\'e duality to recover the stable splitting of compactly supported mapping spaces, $\rm{Map_c}$$(M,\Sigma^nX)$, where $M$ is a parallelizable $n$-manifold. Our method for deriving this splitting is new, and naturally extends to give a more general stable splitting of the space of compactly supported sections of a certain bundle on $M$ with fibers $\Sigma^nX$, twisted by the tangent bundle of $M$. This generalization incorporates possible $O(n)$-actions on $X$ as well as accommodating non-parallelizable manifolds. Read More


We prove that Getzler's higher generalization of the Deligne groupoid commutes with totalization and homotopy limits. Read More


We use descent theoretic methods to solve the homotopy limit problem for Hermitian $K$-theory over very general Noetherian base schemes. As another application of these descent theoretic methods, we compute the cellular Picard group of 2-complete Hermitian $K$-theory over $\mathop{Spec}(\mathbb{C})$, showing that the only invertible cellular spectra are suspensions of the tensor unit. Read More


We use the cobordism category constructed in arXiv:1703.01047 to the study the homotopy type of the space of positive scalar curvature metrics on a spin manifold of dimension > 4. Our methods give an alternative proof and extension of a recent theorem of Botvinnik, Ebert, and Randal-Williams from arXiv:1411. Read More


The paper relates the Gorenstein duality statements studied by the first author to the Anderson duality statements studied by the second author, and explains how to use local cohomology and invariant theory to understand the numerology of shifts in simple cases. Read More


We discuss algorithmic approach to growth of the codimension sequences of varieties of multilinear algebras, or, equivalently, the sequences of the component dimensions of algebraic operads. The (exponentional) generating functions of such sequences are called codimension series of varieties, or generating series of operads. We show that in general there does not exist an algorithm to decide whether the growth exponent of a codimension sequence of a variety defined by given finite sets of operations and identities is equal to a given rational number. Read More


We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0) provided that n is nilpotent on X (resp. Read More


Assuming a conjecture about factorization homology with adjoints, we prove the cobordism hypothesis, after Baez-Dolan, Costello, Hopkins-Lurie, and Lurie. Read More


We produce first examples of p-local height three TAF homology theories. The corresponding one-dimensional formal groups arise as split summands of the formal groups of certain abelian three-folds, the Shimura variety of which can be reinterpreted as moduli of a family of Picard curves. This allows an explicit description of an automorphic form valued genus in terms of the coefficients of these curves. Read More


We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. Read More


Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a nearest neighbor function to smooth the point cloud into a manifold and then apply persistent homology to a Morse filtration. A significant challenge is that this smoothing process involves the choice of a parameter and persistent homology is highly sensitive to that choice; moreover, important scale information is lost. Read More


As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces $X$ and $Y$ certifies that $X$ and $Y$ are approximately isomorphic, a homotopy interleaving between $X$ and $Y$ certifies that $X$ and $Y$ are approximately weakly equivalent. Read More


Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy-Levine-Panin, a theorem of Cisinski-Deglise and a version of the Roendigs-Ostvaer theorem, rational stable motivic homotopy theory over a field of characteristic zero is recovered in this paper from finite Chow-Witt correspondences in the sense of Calmes-Fasel. Read More


For any compact and connected Lie group $G$ and any free abelian or free nilpotent group $\Gamma$ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety) $Rep(\Gamma,G)_1$, with coefficients in a field $F$ with ${char} (F)$ either 0 or relatively prime to the order of the Weyl group $W$. We give explicit formulas for the Poincar\'e series. In addition we study $G$-equivariant stable decompositions of subspaces $X(q,G)$ of the free monoid $J(G)$ generated by the Lie group $G$, obtained from finitely generated free nilpotent group representations. Read More


Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Let C be a category that has a zero object and all small limits. A homogeneous functor (in the sense of manifold calculus) of degree k from O(M) to C is called very good if it sends isotopy equivalences to isomorphisms. Read More


We calculate the mod-two cohomology of all alternating groups together, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. Read More