Theory of self-conjugate cobordisms

The ring of bordisms of manifolds, in the normal bundle to which is given the structure of a self-conjugate complex bundle, is studied. The results of computation of this ring up to dimension nine inclusive are given. The elements of the ring which can be represented by spheres and real projective spaces are considered. Their orders are computed and certain relations between them are found.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 885–896, December, 1977.     

Open conformal cobordisms

Conformal field theories were first axiomatized by Segal (2004) as symmetric monoidal functors from a topological category of conformal cobordisms between compact oriented 1-dimensional manifolds to vector spaces. Costello (2007) later expanded the definition of the category to allow for cobordisms between manifolds with boundaries, and was able to use representations of this category to give a mirror partner for Gromov-Witten invariants. The main goal of this paper is to provide a rigorous definition of the category of open conformal cobordisms. To the best of our knowledge, such a definition does not appear in the literature. Although most results here are probably known to the experts, the proofs are, as far as we can tell, new, and require only elementary results about quasiconformal mappings.     

On symplectic cobordisms

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are “symplectic cobordism equivalent”. Received: 26 March 2001 / Revised version: 1 May 2001 / Published online: 28 February 2002     

The toric cobordisms

We introduce the notions of oriented and unoriented cobordisms in the class of closed 3-manifolds fibered by tori and compute the corresponding cobordism groups.

     

Morse functions on cobordisms

We study the homotopy invariants of crossed and Hilbert complexes. These invariants are applied to the calculation of the exact values of Morse numbers of smooth cobordisms.     

Algebraic cobordisms of a Pfister quadric

In this article we compute the ring of algebraic cobordismsof a Pfister quadric. This is a rare example of a non-cellularvariety where such a computation is known. We consider the algebraiccobordisms * of Levine and Morel, as well as the MGL^{2*, *} ofVoevodsky. The methods of computation in these two cases arequite different. However, the results do agree (which supportsthe expectation that the two theories actually coincide). Weshow that the restriction homomorphism in our case is injectivefor any field extension E/F.     

Imbeddings and homology cobordisms of lens spaces

Both authors partially supported by NSF grants.     

AN INVARIANT FOR HYPERGRAPHS

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An invariant subspace theorem

In this paper it is proved that every operator on a complex Hilbert space whose spectrum is a spectral set has a nontrivial invariant subspace.     

An invariant for unbounded operators

For a class of unbounded operators, a deformation of a Bott projection is used to construct an integer-valued invariant measuring deviation of the non-commutative deformations from the commutative originals, and its interpretation in terms of -theory of -algebras is given. Calculation of this invariant for specific important classes of unbounded operators is also presented.

     

Union and tangle

Shibuya proved that any union of two nontrivial knots without local knots is a prime knot. In this note, we prove it in a general setting. As an application, for any nontrivial knot, we give a knot diagram such that a single unknotting operation on the diagram cannot yield a diagram of a trivial knot.