Intersection homology Künneth theorems

Cohen, Goresky, and Ji showed that there is a Künneth theorem relating the intersection homology groups to and , provided that the perversity satisfies rather strict conditions. We consider biperversities and prove that there is a Künneth theorem relating to and for all choices of and . Furthermore, we prove that the Künneth theorem still holds when the biperversity p, q is “loosened” a little, and using this we recover the Künneth theorem of Cohen–Goresky–Ji.     

The Künneth formula for Hilbert complexes

A variant of the Künneth formula for tensor products of Fredholm complexes of Hilbert spaces is given.     

Rabinowitz–Floer homology for superquadratic Dirac equations on compact spin manifolds

In this paper, we investigate the properties of a semilinear problem on a spin manifold involving the Dirac operator through the construction of Rabinowitz–Floer homology groups. We give several existence results for subcritical and critical nonlinearities as application of the computation of the different homologies.     

Going-down functors, the Künneth formula, and the Baum-Connes conjecture

We study the connection between the Baum-Connes conjecture for a locally compact group G with coeefficient A and the Künneth formula for the K-theory of tensor products by the corresponding crossed product . The main tool for this is obtained by an application of a general reduction procedure which allows us to analyze certain functors connected to the topological K-theory of a group in terms of their restrictions to compact subgroups. We also discuss several other interesting applications of this method, including a general extension result for the Baum-Connes conjecture.     

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.     

Khovanov–Seidel quiver algebras and bordered Floer homology

We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard–Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.     

Instanton Floer homology for connected sums of Poincaré spheres

In this paper, we apply the Mayer-Vietoris principle to compute the integer graded Floer homology of the connected sum of a Poincaré homology sphere with itself. We then apply the Fintushel-Stern spectral sequence to deduce its regular Floer homology and observe the presence of 3-torsion in the Floer groups.Partially supported by NSF grant number DMS-0245323.     

A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology

We study critical points of the indefinite functional by applying Floer's homology construction to the ordinary gradient flow of the functional f on a suitable Sobolev space. One of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the Floer homology for . Received November 11, 1997; in final form March 12, 1998     

Nielsen theory, Floer homology and a generalisation of the Poincaré-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology for symplectomorphisms of surfaces and a calculation of Seidel’s symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and an asymptotic symplectic invariant. A generalisation of the Poincaré-Birkhoff fixed point theorem and Arnold conjecture is proposed. Dedicated to Vladimir Igorevich Arnold     

Hofer’s metric on the space of Lagrangian submanifolds and wrapped Floer homology

In this paper, we study the relationship between wrapped Floer homology and displaceability of a Lagrangian submanifold which we call vanishing theorem of wrapped Floer homology. We also use this theorem to study Hofer’s pseudometric on the space of Lagrangian submanifolds. We prove an inequality, the Lagrangian version of the inequality of Gromov width and displacement energy, which is called energy-capacity inequality.     

Tight Lagrangian homology spheres in compact homogeneous Kähler manifolds

For any irreducible compact homogeneous Kähler manifold, we classify the compact tight Lagrangian submanifolds which have the ?²-homology of a sphere.     

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

A Plücker formula for singular projective varieties

We prove a Plücker formula,for a projective variety X with arbitrary singularities, which expresses the class of X, the degree of the dual variety, in terms of Euler characteristics of X and of two linear sections of X. Moreover, we show that there is no formula whatsoever expressing this degree as a difference of two terms, a deformation invariant and a correction for singularities.     

On another extension of q-Pfaff-Saalschütz formula

In this paper we give an extension of q-Pfaff-Saalschütz formula by means of Andrews-Askey integral. Applications of the extension are also given, which include an extension of q-Chu-Vandermonde convolution formula and some other q-identities.     

Anwendungen der Nielsenschen Kürzungsmethode in Gruppen mit einer definierenden Relation

LetG=a ₁,b ₁, ...,a _q ,b _q | (W ₁ (a ₁,b ₁) ...W _q (a _q b _q ))=1,q1, 2,W _j (a _j ,b _j )1. We solve the isomorphism problem forG inasmuch as we can decide in finitely many steps, if any arbitrary one-relator group is isomorphic toG or not. FurthermoreG turns out to have a finitely generated automorphism group. Forq=1 this was proved byS. J. Pride. Forq2 the proof is based onNielsen's reduction method. There are some other interesting results on subgroup problems for one-relator groups obtained by this method.     

Ueber die Bewegung eines Körpers in einer Flüssigkeit

Ohne Zusammenfassung