scholarly journals On vanishing of all fourfold products of the Ray classes in symplectic cobordism

2020 ◽  
Vol 148 (9) ◽  
pp. 4107-4115
Author(s):  
Malkhaz Bakuradze
Keyword(s):  
Symplectic Cobordism

The symplectic cobordism ring. I

1980 ◽  
Vol 24 (228) ◽  
pp. 0-0 ◽  
Author(s):  
Stanley O. Kochman
Keyword(s):  
Symplectic Cobordism ◽  
Cobordism Ring

scholarly journals Open books and exact symplectic cobordisms

2018 ◽  
Vol 29 (04) ◽  
pp. 1850026 ◽  
Author(s):  
Mirko Klukas
Keyword(s):  
Disjoint Union ◽  
Contact Manifold ◽  
Higher Dimensions ◽  
Fiber Bundles ◽  
Open Book ◽  
Contact Manifolds ◽  
Open Books ◽  
Symplectic Cobordism ◽  
The Given ◽  
Symplectic Cobordisms

Given two open books with equal pages, we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods, we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From this we conclude, following an outline by Wendl, that the complement of the binding of an open book cannot contain any local filling obstruction. Given a contact [Formula: see text]-manifold, according to Eliashberg there is a symplectic cobordism to a fibration over the circle with symplectic fibers. We extend this result to higher dimensions recovering a recent result by Dörner–Geiges–Zehmisch. Our cobordisms can also be thought of as the result of the attachment of a generalized symplectic [Formula: see text]-handle.

scholarly journals Quaternionic Grassmannians and Borel classes in algebraic geometry

2021 ◽  
Vol 33 (1) ◽  
pp. 97-140
Author(s):  
I. Panin ◽  
C. Walter
Keyword(s):  
Algebraic Geometry ◽  
Symplectic Form ◽  
Cell Structure ◽  
Content Type ◽  
Open Subscheme ◽  
Oriented Cohomology ◽  
Symplectic Cobordism ◽  
Sum Formula ◽  
Symplectic Vector Space ◽  
Witt Groups

The quaternionic Grassmannian H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) is the affine open subscheme of the usual Grassmannian parametrizing those 2 r 2r -dimensional subspaces of a 2 n 2n -dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular, we have HP n = H Gr ⁡ ( 1 , n + 1 ) \operatorname {HP}^n = \operatorname {H Gr}(1,n+1) . For a symplectically oriented cohomology theory A A , including oriented theories but also the Hermitian K \operatorname {K} -theory, Witt groups, and algebraic symplectic cobordism, we have A ( HP n ) = A ( pt ) [ p ] / ( p n + 1 ) A(\operatorname {HP}^n) = A(\operatorname {pt})[p]/(p^{n+1}) . Borel classes for symplectic bundles are introduced in the paper. They satisfy the splitting principle and the Cartan sum formula, and they are used to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes. The cell structure of the H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) exists in cohomology, but it is difficult to see more than part of it geometrically. An exception is HP n \operatorname {HP}^n where the cell of codimension  2 i 2i is a quasi-affine quotient of A 4 n − 2 i + 1 \mathbb {A}^{4n-2i+1} by a nonlinear action of G a \mathbb {G}_a .

On the symplectic cobordism ring

1970 ◽  
Vol 45 (1) ◽  
pp. 159-169 ◽  
Author(s):  
D. M. Segal
Keyword(s):  
Symplectic Cobordism ◽  
Cobordism Ring

Singularities and Higher Torsion in Symplectic Cobordism

1994 ◽  
Vol 46 (3) ◽  
pp. 485-516 ◽  
Author(s):  
Boris I. Botvinnik ◽  
Stanley O. Kochman
Keyword(s):  
Spectral Sequence ◽  
Geometric Approach ◽  
Symplectic Cobordism ◽  
Cobordism Ring

AbstractIn this paper we construct higher two-torsion elements of all orders in the symplectic cobordism ring. We begin by constructing higher torsion elements in the symplectic cobordism ring with singularities using a geometric approach to the Adams- Novikov spectral sequence in terms of cobordism with singularities. Then we show how these elements determine particular elements of higher torsion in the symplectic cobordism ring.

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