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preprint2015arXiv

Low dimensional Milnor-Witt stems over R

This article computes some motivic stable homotopy groups over R. For 0 <= p - q <= 3, we describe the motivic stable homotopy groups of a completion of the motivic sphere spectrum. These are the first four Milnor-Witt stems. We start with the known Ext groups over C and apply the rho-Bockstein spectral sequence to obtain Ext groups over R. This is the input to an Adams spectral sequence, which collapses in our low dimensional range.

preprint2015arXiv

The Picard group of topological modular forms via descent theory

This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $\mathbf{E}_{\infty}$ -ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms $TMF$ and the non-periodic and non-connective $Tmf$ . We find that $\mathrm{Pic} (TMF)$ is cyclic of order 576, generated by the suspension $ΣTMF$ (a result originally due to Hopkins), while $\mathrm{Pic}(Tmf) = \mathbb{Z}\oplus \mathbb{Z}/24$ . In particular, we show that there exists an invertible $Tmf$ -module which is not equivalent to a suspension of $Tmf$ .

preprint2015arXiv

Persistent homology and Floer-Novikov theory

We construct "barcodes" for the chain complexes over Novikov rings that arise in Novikov's Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. We moreover prove a continuity result which is a natural analogue both of the classical bottleneck stability theorem in persistent homology and of standard continuity results for spectral invariants, and we use this to prove a C^0-robustness result for the fixed points of Hamiltonian diffeomorphisms. Our approach, which is rather different from the standard methods of persistent homology, is based on a non-Archimedean singular value decomposition for the boundary operator of the chain complex.

preprint2016arXiv

Topological Rigidity for FJ by the Infinite Cyclic Group

We call a group FJ if it satisfies the $K$ - and $L$ -theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$ . We show that if $G$ is FJ, then the simple Borel conjecture (in dimensions $\ge 5$ ) holds for every group of the form $G\rtimes\mathbb Z$ . If in addition $Wh(G\times \mathbb Z)=0$ , which is true for all known torsion free FJ groups, then the bordism Borel conjecture (in dimensions $n\ge 5$ ) holds for $G\rtimes\mathbb Z$ . One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion free group $G$ satisfies the $L$ -theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$ , then any semi-direct product $G\rtimes\mathbb Z$ also satisfies the $L$ -theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$ . Our result is indeed more general and implies the $L$ -theoretic Farrell-Jones conjecture with coefficients in additive categories is closed under extensions of torsion free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.

preprint2016arXiv

Suspension theorems for links and link maps

We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$ ) the group of smooth embeddings $S^p\sqcup S^q\to S^m$ (respectively, $S^p\to S^m$ ) up to smooth isotopy. Denote by $LM^m_{p,q}$ the group of link maps $S^p\sqcup S^q\to S^m$ up to link homotopy. Theorem 1. If $p\le q\le m-3$ and $2p+2q\le 3m-6$ then \begin{equation*} L^m_{p,q}\cong π_p(S^{m-q-1})\oplusπ_{p+q+2-m}(SO/SO_{m-p-1})\oplus C^{m-p}_p\oplus C^{m-q}_q. \end{equation*} Theorem 2. If $p, q\le m-3$ and $2p+2q\le 3m-5$ then $LM^m_{p,q}\cong π^S_{p+q+1-m}$ . Our approach is based on the use of the suspension operation for links and link maps, and suspension theorems for them.

preprint2017arXiv

Cohomology of Effect Algebras

We will define two ways to assign cohomology groups to effect algebras, which occur in the algebraic study of quantum logic. The first way is based on Connes' cyclic cohomology. The resulting cohomology groups are related to the state space of the effect algebra, and can be computed using variations on the Kunneth and Mayer-Vietoris sequences. The second way involves a chain complex of ordered abelian groups, and gives rise to a cohomological characterization of state extensions on effect algebras. This has applications to no-go theorems in quantum foundations, such as Bell's theorem.

preprint2017arXiv

Incompressible fillings of manifolds

We find boundaries of Borel-Serre compactifications of locally symmetric spaces, for which any filling is incompressible. We prove this result by showing that these boundaries have small singular models and using these models to obstruct compressions. We also show that small singular models of boundaries obstruct $S^1$ -actions (and more generally homotopically trivial $\mathbb Z/p$ -actions) on interiors of aspherical fillings. We use this to bound the symmetry of complete Riemannian metrics on such interiors in terms of the fundamental group. We also use small singular models to simplify the proofs of some already known theorems about moduli spaces (the minimal orbifold theorem and a topological analogue of Royden's theorem).

preprint2017arXiv

Vexillary degeneracy loci classes in K-theory and algebraic cobordism

In this paper, we prove determinant formulas for the $K$ -theory classes of the structure sheaves of degeneracy loci classes associated to vexillary permutations in type $A$ . As a consequence we obtain determinant formulas for Lascoux-Schützenberger's double Grothendieck polynomials associated to vexillary permutations. Furthermore, we generalize the determinant formula to algebraic cobordism.

preprint2014arXiv

On the category of props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete and is a symmetric monoidal category, with monoidal product closely related to the Boardman-Vogt tensor product of operads. Tools developed in this article, which is the first part of a larger work, include a generalized version of multilinearity of functors, a free prop construction defined on certain "generalized graphs", and the relationship between the category of props and the categories of permutative categories and of operads.

preprint2016arXiv

Central reflections and nilpotency in exact Mal'tsev categories

We study nilpotency in the context of exact Mal'tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie's functor calculus. We show in particular that the reflection into the subcategory of $n$ -nilpotent objects is the universal endofunctor of degree $n$ if and only if every $n$ -nilpotent object is $n$ -folded. In the special context of a semi-abelian category, an object is $n$ -folded precisely when its Higgins commutator of length $n+1$ vanishes.

preprint2016arXiv

Limit theorems for persistence diagrams

The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the limiting persistence diagram to have the full support. We also discuss a central limit theorem for persistent Betti numbers.

preprint2016arXiv

Involutions on surfaces

We use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, but some questions seem to have been left unanswered. We give a modern treatment that leads to a complete classification.

preprint2016arXiv

Homotopic properties of Kähler orbifolds

We prove the formality and the evenness of odd-degree Betti numbers for compact Kähler orbifolds, by adapting the classical proofs for Kähler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any Kähler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers $b_1=0$ and $b_2\,> 1$ .

preprint2016arXiv

Involutions in the topologists' orthogonal group

We classify conjugacy classes of involutions in the isometry groups of nondegenerate, symmetric bilinear forms over the field of two elements. The new component of this work focuses on the case of an orthogonal form on an even dimensional space. In this context we show that the involutions satisfy a remarkable duality, and we investigate several numerical invariants.

preprint2016arXiv

The First Differential of the Functor "Algebraic K-Theory of Spaces"

In his "Algebraic K-theory of topological spaces II" Waldhausen proved that his functor A(X) splits: There is a canonical map from the stable homotopy of X which has a retraction up to weak equivalence. We adapt Waldhausen's proof to obtain a calculation of the Differential (in the sense of Goodwillie's "Calculus I") of A(X) at any path-connected base space.

preprint2013arXiv

Brauer spaces for commutative rings and structured ring spectra

Using an analogy between the Brauer groups in algebra and the Whitehead groups in topology, we first use methods of algebraic K-theory to give a natural definition of Brauer spectra for commutative rings, such that their homotopy groups are given by the Brauer group, the Picard group and the group of units. Then, in the context of structured ring spectra, the same idea leads to two-fold non-connected deloopings of the spectra of units.

preprint2013arXiv

On the curvature of vortex moduli spaces

We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric powers of the surface), we prove that, for genus g>1, the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kaehler metrics on certain symmetric powers. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.

preprint2015arXiv

Linear extensions and order-preserving poset partitions

We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite $n$ -element poset $P$ with $n\geq 3$ is homotopy equivalent to a wedge of spheres of dimension $n-3$ . If $P$ is connected, then the number of spheres is equal to the number of linear extensions of $P$ . In general, the number of spheres is equal to the number of cyclic extensions of $P$ .

preprint2014arXiv

Nash Equilibria via Duality and Homological Selection

Given a multifunction from $X$ to the $k-$ fold symmetric product $Sym_k(X)$ , we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis of multilinearity. We use basic intersection theory, Poincaré Duality in addition to the Dold-Thom Theorem.

preprint2016arXiv

Strong discrete Morse theory and simplicial L-S category: A discrete version of the Lusternik-Schnirelmann Theorem

We prove a discrete version of the Lusternik-Schnirelmann theorem for discrete Morse functions and the recently introduced simplicial Lusternik-Schnirelmann category of a simplicial complex. To accomplish this, a new notion of critical object of a discrete Morse function is presented, which generalizes the usual concept of critical simplex (in the sense of R. Forman). We show that the non-existence of such critical objects guarantees the strong homotopy equivalence (in the Barmak and Minian's sense) between the corresponding sublevel complexes. Finally, we establish that the number of critical objects of a discrete Morse function defined on $K$ is an upper bound for the non-normalized simplicial Lusternik-Schnirelmann category of $K$ .

preprint2014arXiv

Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups

Let $G$ be a real reductive algebraic group with maximal compact subgroup $K$ , and let $F_r$ be a rank $r$ free group. We show that the space of closed orbits in $\mathrm{Hom}(F_r,G)/G$ admits a strong deformation retraction to the orbit space $\mathrm{Hom}(F_r,K)/K$ . In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups $G$ , such as $\mathrm{Sp}(4,\mathbb{R})$ and $\mathrm{U}(2,2)$ . We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

preprint2016arXiv

Loop near-rings and unique decompositions of H-spaces

For every H-space $X$ the set of homotopy classes $[X,X]$ possesses a natural algebraic structure of a loop near-ring. Albeit one cannot say much about general loop near-rings, it turns out that those that arise from H-spaces are sufficiently close to rings to have a viable Krull--Schmidt type decomposition theory, which is then reflected into decomposition results of H-spaces. In the paper we develop the algebraic theory of local loop near-rings and derive an algebraic characterization of indecomposable and strongly indecomposable H-spaces. As a consequence, we obtain unique decomposition theorems for products of H-spaces. In particular, we are able to treat certain infinite products of H-spaces, thanks to a recent breakthrough in the Krull--Schmidt theory for infinite products. Finally, we show that indecomposable finite $p$ -local H-spaces are automatically strongly indecomposable, which leads to an easy alternative proof of classical unique decomposition theorems of Wilkerson and Gray.

preprint2016arXiv

Homotopical properties of the simplicial Maurer-Cartan functor

We consider the category whose objects are filtered, or complete, $L_\infty$ -algebras and whose morphisms are $\infty$ -morphisms which respect the filtrations. We then discuss the homotopical properties of the Getzler-Hinich simplicial Maurer-Cartan functor which associates to each filtered $L_\infty$ -algebra a Kan simplicial set, or $\infty$ -groupoid. In previous work with V. Dolgushev, we showed that this functor sends weak equivalences of filtered $L_\infty$ -algebras to weak homotopy equivalences of simplicial sets. Here we sketch a proof of the fact that this functor also sends fibrations to Kan fibrations. To the best of our knowledge, only special cases of this result have previously appeared in the literature. As an application, we show how these facts concerning the simplicial Maurer--Cartan functor provide a simple $\infty$ -categorical formulation of the Homotopy Transfer Theorem.

preprint2015arXiv

Minimal fibrations of dendroidal sets

We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for infinity-operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. In an appendix, we explain how our arguments can be used to extend the results of Cisinski, giving the existence of minimal fibrations in model categories of presheaves over generalised Reedy categories of a rather common type. Besides some applications to the theory of algebras over infinity-operads, we also prove a gluing result for parametrized connective spectra (or Gamma-spaces).

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