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We prove a theorem in 3-dimensional topological field theory: a Reshetikhin-Turaev theory admits a nonzero boundary theory iff it is a Turaev-Viro theory. The proof immediately implies a characterization of fusion categories in terms of dualizability. The main theorem applies to physics, where it implies an obstruction to a gapped 3-dimensional quantum system admitting a gapped boundary theory. Appendices on bordism multicategories and on internal duals may be of independent interest.; v2 extensive revision: added theorem on dualizable 2-categories, material on natural transformations, reworked theorems and several proofs, and more., 51 pages, 32 figures; v2 major revision, including new Appendix B; v2 minor changes for publication in Commun. Math. Phys

In this paper, we consider the problem of the nonnegative scalar curvature (NNSC)-cobordism of Bartnik data $$\left( {\sum _1^{n - 1},{{\rm{\gamma }}_1},{H_1}} \right)$$ and $$\left( {\sum _2^{n - 1},{{\rm{\gamma }}_2},{H_2}} \right)$$ . We prove that given two metrics γ1 and γ2 on Sn−1 (3 ⩽ n ⩽ 7) with H1 fixed, then (Sn−1, γ1, H1) and (Sn−1, γ2, H2) admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough (see Theorem 1.3). Moreover, we show that for n = 3, a much weaker condition that the total mean curvature $$\int_{{S^2}}^{} {{H_2}d{\mu _{{\gamma _2}}}} $$ is large enough rules out NNSC-cobordisms (see Theorem 1.2); if we require the Gaussian curvature of γ2 to be positive, we get a criterion for nonexistence of the trivial NNSC-cobordism by using the Hawking mass and the Brown-York mass (see Theorem 1.1). For the general topology case, we prove that $$\left( {\Sigma _1^{n - 1},{{\rm{\gamma }}_1},0} \right)$$ and $$\left( {{\rm{\Sigma }}_2^{n - 1},{{\rm{\gamma }}_2},{H_2}} \right)$$ admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough (see Theorem 1.5).

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produce a framed 3-dimensional local TFT attached to the category of representations of a quantum group at any value of $q$., Comment: Minor updates and edits; final version

In the early 2000s, Ozsv\'{a}th and Szab\'{o} introduced a collection of invariants for 3--manifolds and 4--manifolds called Heegaard Floer homology. To a 3--manifold they constructed a group, and to a 4--manifold which cobounds two 3--manifolds, they constructed a homomorphism between the manifolds appearing on the ends. Their invariants satisfy many of the axioms of a TQFT as described by Atiyah, however their construction has some additional restrictions which prevent it from fitting into Atiyah's framework. There is a refinement of Heegaard Floer homology for 3--manifolds containing a knot, due to Ozsv\'{a}th and Szab\'{o}, and independently Rasmussen, and a further refinement for 3--manifolds containing links, due to Ozsv\'{a}th and Szab\'{o}. It's a natural question as to whether one can define functorial maps associated to link cobordisms.In this thesis, we describe a package of cobordism maps for Heegaard Floer homology and link Floer homology. The cobordism maps satisfy an appropriate analogy of the axiomatic description of a TQFT formulated by Atiyah. To a ribbon graph cobordism between two based 3--manifolds, we associate a map between the Heegaard Floer homologies of the ends. To a decorated link cobordism, we obtain maps on the link Floer homologies of the ends. The maps associated to decorated link cobordisms reduce to the maps for ribbon graphs, in a natural way. As applications, we describe several formulas for mapping class group actions on the Heegaard Floer and knot Floer groups. We prove a new bound on a concordance invariant $\Upsilon_K(t)$ from knot Floer homology, and also see how the link cobordism maps give straightforward proofs of other bounds on concordance invariants from knot Floer homology. We also explore the interaction of the maps with conjugation actions on Heegaard Floer homology and link Floer homology, giving connected sum formulas for involutive Heegaard Floer homology and involutive knot Floer homology.

In this article, we investigate the cobordism maps on periodic Floer homology (PFH). In the first part of the paper, we define the cobordism maps on PFH via Seiberg Witten theory as well as the isomorphism between PFH and Seiberg Witten cohomology. Furthermore, we show that the maps satisfy the holomorphic curve axiom. In the second part of the paper, we give an alternative definition of these maps by using holomorphic curve method, provided that the symplectic cobordisms are Lefschetz fibration satisfying certain nice conditions. Under additionally certain monotonicity assumptions, we show that these two definitions are equivalent., The preview title of this paper is "Cobordism Maps on PFH induced by Lefschetz Fibration over Higher Genus Base". According to referees'suggestion, I change it to be "On cobordism maps on periodic Floer homology". In this version, the exposition of the paper is improved. Besides, I correct some mistakes about the composition rule and restate Theorem 1 for a general setting

In this work, we compute the $0$th cohomology group of a complex of groups of cobordism-framed correspondences, and prove the isomorphism to Milnor $K$-groups. An analogous result for common framed correspondences has been proved by A. Neshitov in his paper "Framed correspondences and the Milnor---Witt $K$-theory". Neshitov's result is, at the same time, a computation of the homotopy groups $\pi_{i,i}(S^0)(Spec(k)).$ This work could be used in the future as basis for computing homotopy groups $\pi_{i,i}(MGL_{\bullet})(Spec(k))$ of the spectrum $MGL_{\bullet}.$, Comment: 18 pages

The homotopy category of the bordism category $hBord_d$ has as objects closed oriented $(d-1)$-manifolds and as morphisms diffeomorphism classes of $d$-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of $hBord_d$ for $d = 1$, exhibiting it as a circle bundle over $\mathbb{CP}^\infty_{-1}$. As part of our proof we construct a quotient $Bord_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space $\Omega^{\infty-2}\mathbb{CP}^\infty_{-1}$ and moreover that, if one equips these bordisms with a map to a simply connected space $X$, the resulting $Bord_1^{red}(X)$ can be thought of as a cobordism model for the topological cyclic homology $TC(\mathbb{S}[\Omega X])$. In the second part of the paper we construct an infinite loop space map $B(hBord_1^{red}) \to Q(\Sigma^2 \mathbb{CP}^\infty_+)$ in this model and use it to derive combinatorial formulas for rational cocycles on $Bord_1^{red}$ representing the Miller-Morita-Mumford classes $\kappa_i \in H^{2i+2}((B(hBord_1); \mathbb{Q})$., Comment: 46 pages, 9 figures. Final version, to appear in the Journal of Topology

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $E_\infty$-ring spectra in various ways. In this work we first establish, in the context of $\infty$-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $E_\infty$-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let $R$ be an $E_\infty$-ring spectrum and $\mathrm{Pic}(R)$ denote its Picard $E_\infty$-group. Let $Mf$ denote the Thom $E_\infty$-$R$-algebra of a map of $E_\infty$-groups $f:G\to \mathrm{Pic}(R)$; examples of $Mf$ are given by various flavors of cobordism spectra. We prove that the cotangent complex of $R\to Mf$ is equivalent to the smash product of $Mf$ and the connective spectrum associated to $G$., Comment: 22 pages. Final version

Kronheimer and Mrowka introduced a new knot invariant, called $s^\sharp$, which is a gauge theoretic analogue of Rasmussen's $s$ invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including algebraic knots and the connected sums of quasi-positive knots with non-trivial right handed torus knots. These computations reveal some unexpected phenomena: we show that $s^\sharp$ does not have to agree with $s$, and that $s^\sharp$ is not additive under connected sums of knots. Inspired by our computations, we separate the invariant $s^\sharp$ into two new invariants for a knot $K$, $s^\sharp_+(K)$ and $s^\sharp_-(K)$, whose sum is $s^\sharp(K)$. We show that their difference satisfies $0 \leq s^\sharp_+(K) - s^\sharp_-(K) \leq 2$. This difference may be of independent interest. We also construct two link concordance invariants that generalize $s^\sharp_\pm$, one of which we continue to call $s^\sharp_\pm$, and the other of which we call $s^\sharp_I$. To construct these generalizations, we give a new characterization of $s^\sharp$ using immersed cobordisms rather than embedded cobordisms. We prove some inequalities relating the genus of a cobordism between two links and the invariant $s^\sharp$ of the links. Finally, we compute $s^\sharp_\pm$ and $s^\sharp_I$ for torus links.

Let M be a closed smooth manifold. In 1999, Friedlander and Nadirashvili introduced a new differential invariant $$I_1(M)$$ using the first normalized nonzero eigenvalue of the Lalpace–Beltrami operator $$\Delta _g$$ of a Riemannian metric g. They defined it taking the supremum of this quantity over all Riemannian metrics in each conformal class, and then taking the infimum over all conformal classes. By analogy we use k-th eigenvalues of $$\Delta _g$$ to define the invariants $$I_k(M)$$ indexed by positive integers k. In the present paper the values of these invariants on surfaces are investigated. We show that $$I_k(M)=I_k({\mathbb {S}}^2)$$ unless M is a non-orientable surface of even genus. For orientable surfaces and $$k=1$$ this was earlier shown by Petrides. In fact Friedlander and Nadirashvili suggested that $$I_1(M)=I_1({\mathbb {S}}^2)$$ for any surface M different from $${\mathbb {RP}}^2$$ . We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has $$I_k(M)>I_k({\mathbb {S}}^2)$$ . We also discuss the connection between the Friedlander–Nadirashvili invariants and the theory of cobordisms, and conjecture that $$I_k(M)$$ is a cobordism invariant.

For a closed oriented smooth 4–manifold X with b+2(X)>0, the Seiberg–Witten invariants are well-defined. Taubes’ “ SW= Gr” theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes’ Gromov invariants. In the absence of a symplectic form, there are still nontrivial closed self-dual 2–forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describe well-defined integral counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2–forms, and it is shown that they recover the Seiberg–Witten invariants over ℤ∕2ℤ. This is an extension of “ SW= Gr” to nonsymplectic 4–manifolds. The main result of this paper asserts the following. Given a suitable near-symplectic form ω and tubular neighborhood 𝒩 of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X−𝒩,ω) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form “near-symplectic” Gromov invariants as a function of spin-c structures on X.

Cobordism of virtual string links on n strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links on n strands up to cobordisms and elements of the direct product of n ( n − 1) copies of the integers. This paper also shows that virtual string links up to unwelded equivalence are classified by those groups. Finally, the related theory of welded string link cobordism is defined herein and shown to be trivial for string links with one component.

We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either $\mathbb{Z}_2$- or $(\mathbb{Z}_2 \times \mathbb{Q})$-graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object $W \in \Bbbk[x_1,\dots,x_n]$ determines a framed extended TQFT. We then compute the Serre automorphisms $S_W$ to show that $W$ determines an oriented extended TQFT if the associated category of matrix factorisations is $(n-2)$-Calabi-Yau. The extended TQFTs we construct from $W$ assign the non-separable Jacobi algebra of $W$ to a circle. This illustrates how non-separable algebras can appear in 2-1-0-extended TQFTs, and more generally that the question of extendability depends on the choice of target category. As another application, we show how the construction of the extended TQFT based on $W=x^{N+1}$ given by Khovanov and Rozansky can be derived directly from the cobordism hypothesis., Comment: 29 pages

The contact invariant is an element in the monopole Floer homology groups of an oriented closed three manifold canonically associated to a given contact structure. A non-vanishing contact invariant implies that the original contact structure is tight, so understanding its behavior under symplectic cobordisms is of interest if one wants to further exploit this property. By extending the gluing argument of Mrowka and Rollin to the case of a manifold with a cylindrical end, we will show that the contact invariant behaves naturally under a strong symplectic cobordism. As quick applications of the naturality property, we give alternative proofs for the vanishing of the contact invariant in the case of an overtwisted contact structure, its non-vanishing in the case of strongly fillable contact structures and its vanishing in the reduced part of the monopole Floer homology group in the case of a planar contact structure. We also prove that a strong filling of a contact manifold which is an L-space must be negative definite., More details added. To appear in Algebraic & Geometric Topology

The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $n \geq 2$, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic $n$-manifolds. An analogous result holds for non-compact orientable hyperbolic arithmetic $n$-manifolds of finite volume that are geometric boundaries for $n \geq 2$.

In this paper, we introduce a geometric technique of working with skew-framed manifolds. It allows us to study stable homotopy groups of some Thom spaces by geometric means. We schematically describe how our results (which are also of independent interest) can be applied to obtain a proof of the Baum–Browder theorem stating nonimmersibility of ℝP10 to ℝ15.

We give an explicit construction of the Honda--Kazez--Mati\'c gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree., Comment: 86 pages, 54 figures, to appear in Geometry and Topology

Изучается понятие „intermediate bordism group" , которое было введено П. Дж. Экклзом для исследовании фильтраций в стабильных гомотопических группах сфер. Введено новое понятие группы кобордизма стабильно-оснащенных погружений. Строится представляю щее пространство для новых групп и вычисляются ранги этих групп кобордизма. Инварианты Хопфа и гомоморфизм Кана-Придди обобщаются на группы кобордизма стабильно-оснащенных погружений.

We compute the homology of the spaces in the Omega spectrum for $BoP$. There is no torsion in $H_*(\underline{BoP}_{\; i})$ for $i \ge 2$, and things are only slightly more complicated for $i 6$. This completes the computation of all $H_*(\underline{MSU}_{\;*})$.

We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted and has self-linking number -1. The same is true moreover for any contact structure on a closed 3-manifold that is reducible. Our results generalize an earlier theorem of Hofer-Wysocki-Zehnder for the 3-sphere, but use somewhat newer techniques: the main idea is to exploit the intersection theory of punctured holomorphic curves in order to understand the compactification of the space of so-called "nicely embedded" curves in symplectic cobordisms. In the process, we prove a local adjunction formula for holomorphic annuli breaking along a Reeb orbit, which may be of independent interest., 50 pages, 5 figures; v3 implements minor changes suggested by the referees; to appear in J. Symplectic Geom

Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that knot is smoothly concordant to a knot which is homotopic to a smoothly slice knot. As a consequence, we prove that the equivalence relation on knots in homology spheres given by cobounding immersed annuli in a homology cobordism is generated by concordance in homology cobordisms together with homotopy in a homology sphere., Comment: 10 pages, 1 figure. Changes from Version 1: Theorem 1.6 from version 1 was previously proven by the same technique by Austin-Rolfsen. The result of Theorem 1.9 frmo version 1 appears in a remark of Daemi

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the semisimple case. More generally, we characterize invertibility for E_1- and E_2-algebras in an arbitrary symmetric monoidal oo-category, and we conjecture a similar characterization of invertible E_n-algebras for any n. Finally, we propose the Picard group of BrTens as a generalization of the Witt group of non-degenerate braided fusion categories, and pose a number of open questions about it.

In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space $M_\Sigma$ with singular stratum $\beta M$ (a closed manifold of positive codimension) and associated link equal to $L$, a smooth compact manifold. We briefly call such spaces manifolds with $L$-fibered singularities. Under suitable spin assumptions we give necessary index-theoretic conditions for the existence of wedge metrics of positive scalar curvature. Assuming in addition that $L$ is a simply connected homogeneous space of positive scalar curvature, $L=G/H$, with the semisimple compact Lie group $G$ acting transitively on $L$ by isometries, we investigate when these necessary conditions are also sufficient. Our main result is that our conditions are indeed sufficient for large classes of examples, even when $M_\Sigma$ and $\beta M$ are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.

Let $L \subset \mathbb R \times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold $M$. Assume that $L$ has cylindrical Legendrian ends $\Lambda_\pm \subset J^1(M)$. It is well known that the Legendrian contact homology of $\Lambda_\pm$ can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent bundle of $M$. It is also known that this count can be lifted to a mod 2 count of pseudo-holomorphic disks in the symplectization $\mathbb R \times J^1(M)$, and that $L$ induces a morphism between the $\mathbb Z_2$-valued DGA:s of the ends $\Lambda_\pm$ in a functorial way. We prove that this hold with integer coefficients as well. The proofs are built on the technique of orienting the moduli spaces of pseudo-holomorphic disks using capping operators at the Reeb chords. We give an expression for how the DGA:s change if we change the capping operators., Comment: 41 pages, final version, accepted for publication in Quantum Topology. More details have been added to some of the proofs

Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T * Q is an h-cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

Connected sums of lens spaces which smoothly bound a rational homology ball are classified by P Lisca. In the classification, there is a phenomenon that a connected sum of a pair of lens spaces [math] appears in one of the typical cases of rational homology cobordisms. We consider smooth negative-definite cobordisms among several disjoint union of lens spaces and a rational homology [math] –sphere to give a topological condition for the cobordism to admit the above “pairing” phenomenon. By using Donaldson theory, we show that if [math] has a certain minimality condition concerning the Chern–Simons invariants of the boundary components, then any [math] must have a counterpart [math] in negative-definite cobordisms with a certain condition only on homology. In addition, we show an existence of a reducible flat connection through which the pair is related over the cobordism. As an application, we give a sufficient condition for a closed smooth negative-definite [math] –orbifold with two isolated singular points whose neighborhoods are homeomorphic to the cones over lens spaces [math] and [math] to admit a finite uniformization.

Let M be a 2n-dimensional closed unitary manifold with a Tn−1-action with only isolated fixed points. In this paper, we first prove that the equivariant cobordism class of a unitary Tn−1-manifold M is just determined by the equivariant Chern numbers $$c_\omega ^{{T^{n - 1}}}$$ [M], where ω = (i1, i2, ..., i6) are the multi-indexes for all i1, i2, ..., i6 ∈ ℕ. Then we show that if M does not bound equivariantly, then the number of fixed points is greater than or equal to ⌈n/6⌉ + 1, where ⌈n/6⌉ denotes the minimum integer no less than n/6.

We consider seriously the analogy between interpolation of nonlinear functions and manifold learning from samples, and examine the results of transferring ideas from each of these domains to the other. Illustrative examples are given in approximation theory, variational calculus (closed geodesics), and quantitative cobordism theory.

A smooth stably complex manifold is said to be totally tangentially/normally split if its stably tangential/normal bundle is isomorphic to a sum of complex line bundles. It is proved that each class of degree greater than 2 in the graded unitary cobordism ring contains a quasitoric totally tangentially and normally split manifold.

Made available in DSpace on 2019-10-04T12:40:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-01-01 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) BREUDS European Programme The aim of this work is to present some remarks on cobordism of normally nonsingular maps between compact locally orientable Witt spaces. By using the Wu classes defined by Goresky and Pardon, we give a definition of Stiefel-Whitney numbers in this situation. Following Stong's method, we construct a map in the respective intersection homology groups and show that null-cobordism implies the vanishing of these Stiefel-Whitney numbers. Aix Marseille Univ, CNRS, Dept Math, I2M, Marseille, France Sao Paulo State Univ, Dept Math, Rio Claro, Brazil Univ Sao Paulo, Dept Math, Sao Carlos, SP, Brazil Sao Paulo State Univ, Dept Math, Rio Claro, Brazil FAPESP: 2015/06697-9 FAPESP: 2016/24707-4 FAPESP: 2014/00304-2 CNPq: 400580/2012-8 CNPq: 301495/2017-3 CNPq: 482183/2013-6 CNPq: 300733/2009-7 CNPq: 421440/2016-3

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $\Upsilon_K(t)$ for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, closed surface in $S^4$ are determined by the genus of the surface. We also prove a new adjunction relation and adjunction inequality for the link cobordism maps. Along the way, we see how many known results in Heegaard Floer homology can be proven using basic properties of the link cobordism maps, together with the grading change formula.

In this extended note we give a precise definition of fully extended topological field theories \`a la Lurie. Using complete n-fold Segal spaces as a model, we construct an (∞,n)-category of n-dimensional cobordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of cobordisms nCob and the cobordism bicategory nCobext from it.

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G \to H$ of finite groups assigns in a functorial way to a $G$-equivariant topological field theory an $H$-equivariant topological field theory, the pushforward theory. When $H$ is the trivial group, this yields an orbifold construction for $G$-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization., 21 pages, accepted for publication in the Journal of Pure and Applied Algebra

We extend parts of the Lagrangian spectral invariants package recently developed by Leclercq and Zapolsky to the theory of Lagrangian cobordism developed by Biran and Cornea. This yields a nondegenerate Lagrangian "spectral metric" which bounds the Lagrangian "cobordism metric" (recently introduced by Cornea and Shelukhin) from below. It also yields a new numerical Lagrangian cobordism invariant as well as new ways of computing certain asymptotic Lagrangian spectral invariants explicitly., Comment: 25 pages, 1 figure, v3: Major revision. Added new results on non-degeneracy of spectral metric. Final version. To appear in Journal of Topology and Analysis

The purpose of this article is to show that the bivariant algebraic $A$-cobordism groups considered previously by the author are independent of the chosen base ring $A$. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner--Floyd theorem and the Grothendieck--Riemann--Roch theorem to this setting. The general definitions of bivariant cobordism is based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work., Minor modifications

This paper shows that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$, Appendix H], where $G$ is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber--Panov--Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen's geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of $({\Bbb Z}_2)^k$-equivariant unoriented bordism and can still derive the classical result of tom Dieck., Comment: 13 pages. This is the new version to cover the previous version "Equivariant unitary bordism and equivariant cohomology Chern numbers" in 2014

We prove that the tangent bundle of a generic space of planar n-gons with specified side lengths, identified under isometry, plus a trivial line bundle is isomorphic to (n-2) times a canonical line bundle. We then discuss consequences for orientability, cobordism class, immersions, and parallelizability., Replaces an earlier version called "The tangent bundle of planar polygon spaces." Adds a good result about parallelizability

For any given graph G, consider the graph Ĝ which is a cone over G. We study two important invariants of such a cone, namely, the complexity (the number of spanning trees) and the Jacobian of the graph. We prove that complexity of graph Ĝ coincides with the number of rooted spanning forests in G and the Jacobian of Ĝ is isomorphic to the cokernel of the operator I + L(G), where L(G) is the Laplacian of G and I is the identity matrix. As a consequence, one can calculate the complexity of Ĝ as det(I + L(G)). As an application, we establish general structural theorems for the Jacobian of Ĝ in the case when G is a circulant graph or cobordism of two circulant graphs., Журнал «Математические заметки СВФУ», Выпуск 2 (110) 2021, Pages 88-101

This note answers a question of V. V. Vershinin concerning the properties of Buchstaber's elements $\theta_{2i+1} (2)$ in the symplectic cobordism ring of the real projective plane. It is motivated by Roush's famous result that the restriction of these elements to the projective line is trivial, and by the relationship with obstructions to multiplication in symplectic cobordism with singularities.

We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums., 12 pages. To appear in J. Inst. Math. Jussieu

We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K -theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all surfaces and dimension 1 quotients for surfaces X with p g > 0 . We also show that the generating series of virtual cobordism classes can be irrational. Given a K -theory class on X of rank r , we associate natural series of virtual Segre and Verlinde numbers. We show that the Segre and Verlinde series match in the following cases: • [(i)] Quot schemes of dimension 0 quotients, • [(ii)] Hilbert schemes of points and curves over surfaces with p g > 0 , • [(iii)] Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes. Moreover, for punctual quotients of the trivial sheaf of rank N , we prove a new symmetry of the Segre/Verlinde series exchanging r and N . The Segre/Verlinde statements have analogues for punctual Quot schemes over curves.

In this paper we study higher torsion in the symplectic cobordism ring. We use Toda brackets and manifolds with singularities to construct elements of higher torsion and use the Adams spectral sequence to determine an upper bound for the order of these elements.

Gay and Kirby introduced the notion of a trisection of a smooth 4-manifold, which is a decomposition of the 4-manifold into three ekementary pieces. Rubinstein and Tillmann later extended this idea to construct multisections of piecewise-linear manifolds in all dimensions. Given a PL manifold Y of dimension n, this is a decomposition of Y into \(\left\lfloor {{\text{n}}/{2}} \right\rfloor + {1}\) PL submanifolds. We show that every smooth, oriented, compact 5-manifold admits a smooth trisection. Furthermore, given a smooth cobordism W between trisected 4-manifolds, there is a smooth trisection of W extending the trisections on its boundary.

For each integral homology sphere $Y$, a function $\Gamma_Y$ on the set of integers is constructed. It is established that $\Gamma_Y$ depends only on the homology cobordism of $Y$ and it recovers the Fr{\o}yshov invariant. A relation between $\Gamma_Y$ and Fintushel-Stern's $R$-invariant is stated. It is shown that the value of $\Gamma_Y$ at each integer is related to the critical values of the Chern-Simons functional. Some topological applications of $\Gamma_Y$ are given. In particular, it is shown that if $\Gamma_Y$ is trivial, then there is no simply connected homology cobordism from $Y$ to itself., Comment: 44 pages. Comments are welcome! v2: Remark 1.6 (due to Levin and Lidman) added to the paper, references updated

We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks where the $i$-morphisms are given by $i$-fold coisotropic correspondences. Assuming an expected equivalence of different models of higher Morita categories, we prove that all derived Poisson stacks are fully dualizable, and so determine framed extended TQFTs by the Cobordism Hypothesis. Along the way we also prove that the higher Morita category of $E_{n}$-algebras with respect to coproducts is equivalent to the higher category of iterated cospans., Comment: 51 pages, v2: accepted version

We refine prior bounds on how the multivariable signature and the nullity of a link change under link cobordisms. The formula generalizes a series of results about the 4-genus having their origins in the Murasugi-Tristram inequality, and at the same time extends previously known results about concordance invariance of the signature to a bigger set of allowed variables. Finally, we show that the multivariable signature and nullity are also invariant under $0.5$-solvable cobordism., Comment: 41 pages, 3 figures

The purpose of this paper is to describe a dictionary geometry algebra in Lagrangian topology. As a by-product we obtain a tautological (in a sense explained in the body of the paper) proof of a folklore conjecture (sometimes attributed to Kontsevich) claiming that the objects and structure of the derived Fukaya category can be represented through immersed Lagrangians. Our construction is based on certain Lagrangian cobordism categories endowed with a structure called "surgery models"., Comment: 53 figures

In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms., Comment: 61 pages, 17 figures. Final version, accepted for publication in Journal of Differential Geometry, some missing citations have been added