scholarly journals Counting saddle connections in a homology class modulo \begin{document}$ \boldsymbol q $\end{document} (with an appendix by Rodolfo Gutiérrez-Romo)

2019 ◽  
Vol 15 (0) ◽  
pp. 237-262
Author(s):  
Michael Magee ◽  
◽  
Rene Rühr ◽  
Keyword(s):  
Homology Class

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

scholarly journals The Baum-Connes conjecture, noncommutative Poincaré duality, and the boundary of the free group

2003 ◽  
Vol 2003 (38) ◽  
pp. 2425-2445 ◽  
Author(s):  
Heath Emerson
Keyword(s):  
Free Group ◽  
Homology Class ◽  
Hyperbolic Group ◽  
Cross Product ◽  
Direct Connection ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Duality Map ◽  
The Cross ◽  
Gromov Boundary

For every hyperbolic groupΓwith Gromov boundary∂Γ, one can form the cross productC∗-algebraC(∂Γ)⋊Γ. For each such algebra, we construct a canonicalK-homology class. This class induces a Poincaré duality mapK∗(C(∂Γ)⋊Γ)→K∗+1(C(∂Γ)⋊Γ). We show that this map is an isomorphism in the case ofΓ=𝔽2, the free group on two generators. We point out a direct connection between our constructions and the Baum-Connes conjecture and eventually use the latter to deduce our result.

scholarly journals On Analytic Todd Classes of Singular Varieties

2019 ◽  
Author(s):  
Francesco Bei ◽  
Paolo Piazza
Keyword(s):  
Fundamental Group ◽  
Complex Space ◽  
Homology Class ◽  
Classifying Space ◽  
Projective Varieties ◽  
Singular Varieties ◽  
Push Forward ◽  
Birational Invariant ◽  
Complex Projective

Abstract Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ with $\pi :M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth }_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial }+\overline{\partial }^t$ on $M$. In the 2nd part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini–Study metric. First, assuming $\dim (V)\leq 2$, we compare the Baum–Fulton–MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial }$ complex. We show that there is no $L^2$-$\overline{\partial }$ complex on $(\operatorname{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum–Fulton–MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

scholarly journals Invariants of Legendrian Knots in Circle Bundles

2003 ◽  
Vol 05 (04) ◽  
pp. 569-627 ◽  
Author(s):  
Joshua M. Sabloff
Keyword(s):  
Contact Structure ◽  
Homology Class ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Circle Bundle ◽  
Differential Graded Algebra ◽  
Circle Bundles ◽  
Standard Contact ◽  
Definition Of

Let M be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in M. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using "Morse–Bott" techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a nontrivial homology class, but are not Legendrian isotopic.

scholarly journals The Homology Class of a Poisson Transversal

2018 ◽  
Vol 2020 (10) ◽  
pp. 2952-2976
Author(s):  
Pedro Frejlich ◽  
Ioan Mărcuț
Keyword(s):  
Homology Class ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Symplectic Case

Abstract This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent nontrivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Finally, we extend our results to Dirac geometry.

scholarly journals On Realising Mod-2 Homology Classes of Manifolds by Submanifolds

1979 ◽  
Vol 22 (2) ◽  
pp. 109-111 ◽  
Author(s):  
Stavros Papastavridis
Keyword(s):  
Homology Class ◽  
Higher Dimension ◽  
Fundamental Paper

In his fundamental paper (3), Thorn proved, among other things, that a mod-2 homology class of an n-dimensional, closed, compact, C∞ manifold, which has dimension n/2, can be realised by a submanifold, (see (3), Théorème II. 1 and Corollaire 11.13).In this note we examine the question of realisability of mod-2 homology classes of the next higher dimension.

On Immersion of Manifolds

1960 ◽  
Vol 12 ◽  
pp. 529-534 ◽  
Author(s):  
Hans Samelson
Keyword(s):  
Euclidean Space ◽  
Tangent Bundle ◽  
Normal Bundle ◽  
Vector Fields ◽  
Elementary Proof ◽  
Homology Class ◽  
Oriented Manifold ◽  
Normal Degree ◽  
Special Case

In (3) R. Lashof and S. Smale proved among other things the following theorem. If the compact oriented manifold M is immersed into the oriented manifold M', with dim M' ≥ dim M + 2, then the normal degree of the immersion is equal to the Euler-Poincaré characteristic x of M reduced module the characteristic x’ of M'. If M’ is not compact, x' is replaced by 0. “Manifold” always means C∞-manifold. An immersion is a differentiable (that is, C∞) map f whose differential df is non-singular throughout. The normal degree is defined in a certain fashion using the normal bundle of M in M', derived from f, and injecting it into the tangent bundle of M'It is our purpose to give an elementary proof, using vector fields, of this theorem, and at the same time to identify the homology class that represents the normal degree (Theorem I), and to give a proof, using the theory of Morse, for the special case M’ = Euclidean space (Theorem II).

Chains that realize the Gromov invariant of hyperbolic manifolds

1997 ◽  
Vol 17 (3) ◽  
pp. 643-648 ◽  
Author(s):  
DOUGLAS JUNGREIS
Keyword(s):  
Hyperbolic Manifold ◽  
Homology Class ◽  
Hyperbolic Manifolds ◽  
Uniform Measure ◽  
Geodesic Boundary ◽  
Maximal Volume ◽  
Totally Geodesic ◽  
Gromov Norm

For any closed hyperbolic manifold of dimension $n \geq 3$, suppose a sequence of $n$-cycles representing the fundamental homology class have norms converging to the Gromov invariant. We show that this sequence must converge to the uniform measure on the space of maximal-volume ideal simplices. As a corollary, we show that for a hyperbolic $n$-manifold $L$ ($n \geq 3$) with totally-geodesic boundary, the Gromov norm of ($L,\partial L$) is strictly greater than the volume of $L$ divided by the maximal volume of an ideal $n$-simplex.

scholarly journals HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

2010 ◽  
Vol 02 (01) ◽  
pp. 1-55 ◽  
Author(s):  
JEAN-PAUL BRASSELET ◽  
JÖRG SCHÜRMANN ◽  
SHOJI YOKURA
Keyword(s):  
Chern Class ◽  
Blow Up ◽  
Homology Class ◽  
Grothendieck Group ◽  
Characteristic Classes ◽  
Chern Classes ◽  
Singular Spaces ◽  
Coherent Sheaves ◽  
Roch Theorem ◽  
Mixed Hodge Modules

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformationmCy: K0( var /X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0( var /X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation Ty* : K0( var /X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty* is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1). We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe. All our results can be extended to varieties over a base field k of characteristic 0.

scholarly journals HOMOLOGOUS NON-ISOTOPIC SYMPLECTIC TORI IN A K3 SURFACE

2005 ◽  
Vol 07 (03) ◽  
pp. 325-340 ◽  
Author(s):  
TOLGA ETGÜ ◽  
B. DOUG PARK
Keyword(s):  
Homology Class ◽  
Infinite Family ◽  
Infinite Subset ◽  
K3 Surface ◽  
Elliptic Surfaces

For each member of an infinite family of homology classes in the K3 surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also explain how these tori can be non-isotopically embedded as homologous symplectic submanifolds in many other symplectic 4-manifolds including the elliptic surfaces E(n) for n > 2.

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