Piunikhin-Salamon-Schwarz isomorphisms for Lagrangian intersections

We study the intersections of gradient trajectories and holomorphic discs with Lagrangian boundary conditions in cotangent bundles, and give a construction of Piunikhin-Salamon-Schwarz isomorphisms in Lagrangian intersections Floer homology.     

Elliptic curve configurations on Fano surfaces

The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces.     

Floer Homology and the Heat Flow

We study the heat flow in the loop space of a closed Riemannian manifold M as an adiabatic limit of the Floer equations in the cotangent bundle. Our main application is a proof that the Floer homology of the cotangent bundle, for the Hamiltonian function kinetic plus potential energy, is naturally isomorphic to the homology of the loop space. J.W. received partial financial support from TH-Projekt 00321. Received: December 2004 Revision: September 2005 Accepted: September 2005     

The Growth Rate of Symplectic Homology and Affine Varieties

We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.     

A cotangent fibre generates the Fukaya category

We prove that the algebra of chains on the based loop space recovers the derived (wrapped) Fukaya category of the cotangent bundle of a closed smooth oriented manifold. The main new idea is the proof that a cotangent fibre generates the Fukaya category using a version of the map from symplectic cohomology to the homology of the free loop space introduced by Cieliebak and Latschev.     

The homology of path spaces and Floer homology with conormal boundary conditions

We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of the natural path space associated to the boundary conditions. Dedicated to Felix E. Browder     

Filtrations on the knot contact homology of transverse knots

We construct a new invariant of transverse links in the standard contact structure on ${\mathbb R }^3.$ This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link, see (Ekholm et al., Knot contact homology, Arxiv:1109.1542, 2011; Ng, Duke Math J 141(2):365–406, 2008). Here the knot contact homology of a link in ${\mathbb R }^3$ is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle $S^*{\mathbb R }^3$ of ${\mathbb R }^3$ , and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.     

Corrigendum: On the Floer Homology of Cotangent Bundles

We fix an orientation issue that appears in our previous paper about the isomorphism between Floer homology of cotangent bundles and loop space homology. When the second Stiefel‐Whitney class of the underlying manifold does not vanish on 2‐tori, this isomorphism requires the use of a twisted version of the Floer complex. © 2014 Wiley Periodicals, Inc.     

Models for operads

We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps including a homology isomorphism. This approach, naturally arising in string theory, leads us to consider various versions of models. Some applications in topology (homotopy-everything spaces), algebra (cotangent cohomology) and mathematical physics (closed string-field theory) - are also given     

Hochschild homology and split pairs

We study the Hochschild homology of algebras related via split pairs, and apply this to fiber products, trivial extensions, monomial algebras, graded-commutative algebras and quantum complete intersections. In particular, we compute lower bounds for the dimensions of both the Hochschild homology and cohomology groups of quantum complete intersections.     

Orbital variety closures and the convolution product in Borel–Moore homology

This paper settles a twenty-year-old conjecture describing the inclusion relations between orbital variety closures. The solution is in terms of the top Borel–Moore homology of the Steinberg variety and mirrors the way in which the Verma module multiplicities determine the inclusion relations of primitive ideals. It thus gives a link between geometry and representation theory which is more precise than what one would obtain by a naive application of the orbit method. Unlike the primitive ideal case which uses Duflo involutions, the geometric result exploits a link between correspondences and the moment map pertaining to the cotangent bundle on the flag variety. Krull equidimensionality is needed to ensure that all correspondences are recovered from the homology convolution product.     

Floer–Novikov homology and applications to Lagrangian submanifolds

We use a non-Hamiltonian version of Lagrangian Floer homology to prove that an exact Lagrangian submanifold in the cotangent bundle of the 3-torus T ³ must be diffeomorphic to T ³. This improves a previous result of Fukaya, Seidel and Smith.     

A nondisplaceable Lagrangian torus in T*S2

We show that the Lagrangian torus in the cotangent bundles of the 2‐sphere obtained by applying the geodesic flow to the unit circle in a fiber is not displaceable by computing its Lagrangian Floer homology. The computation is based on a symmetry argument. © 2007 Wiley Periodicals, Inc.     

Exact Lagrangians in plumbings

Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Spin Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A ₂-Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any compactly supported symplectomorphism.     

Morse homology for the heat flow

We use the heat flow on the loop space of a closed Riemannian manifold—viewed as a parabolic boundary value problem for infinite cylinders—to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined by counting, modulo time shift, heat flow trajectories between geodesics of Morse index difference one. By Salamon and Weber (GAFA 16:1050–138, 2006) this heat flow homology is naturally isomorphic to Floer homology of the cotangent bundle for Hamiltonians given by kinetic plus potential energy.     

Global Hochschild (co-)homology of singular spaces

We introduce Hochschild (co-)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so-called (derived) Hochschild complex of a morphism; the Hochschild cohomology and homology groups are then the Ext and Tor groups of that complex. We prove that these objects are well defined, extend the known cases, and have the expected functorial and homological properties such as graded commutativity of Hochschild cohomology and existence of the characteristic homomorphism from Hochschild cohomology to the (graded) centre of the derived category.     

On the Floer homology of cotangent bundles

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L^∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W^1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.     

Produit tensoriel et complexe cotangent

With two commutative A-algebras B and C, let us consider their tensor product D and a D-module W. We define and study homology modules H_n (A, B, C, W) allowing the comparison of the homology modules H_n (A, B, W) and H_n (C, D, W) of the theory of cotangen complex. For small n the new modules H_n (A, B, C, W) can be computed in some way. Many explicit results (old and new) of the theory of cotangent complex are consequences of such computations.      

Floer homology for magnetic fields with at most linear growth on the universal cover

The Floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold, as proved by Abbondandolo and Schwarz, Salamon and Weber, and Viterbo. In this paper we show that in the presence of a Dirac magnetic monopole which admits a primitive with at most linear growth on the universal cover, the Floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology. As a consequence we prove that for any atoroidal free homotopy class and any sufficiently small $\tau >0$, any magnetic flow associated to the Dirac magnetic monopole has a closed orbit of period τ belonging to the given free homotopy class. In the case where the Dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the Conley conjecture for Hamiltonians that are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.