scholarly journals The Weinstein Conjecture for Connected Sums

2015 ◽  
pp. rnv124
Author(s):  
Hansjörg Geiges ◽  
Kai Zehmisch
Keyword(s):  
Connected Sums ◽  
Weinstein Conjecture

Quasipositive Links and Connected Sums

2020 ◽  
Vol 54 (1) ◽  
pp. 64-67
Author(s):  
S. Yu. Orevkov
Keyword(s):  
Connected Sums

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals The four-genus of connected sums of torus knots

2017 ◽  
Vol 164 (3) ◽  
pp. 531-550
Author(s):  
CHARLES LIVINGSTON ◽  
CORNELIA A. VAN COTT
Keyword(s):  
Linear Combinations ◽  
Torus Knots ◽  
Connected Sums ◽  
The Third

AbstractWe study the four-genus of linear combinations of torus knots:g4(aT(p, q) #-bT(p′, q′)). Fixing positivep, q, p′, andq′, our focus is on the behavior of the four-genus as a function of positiveaandb. Three types of examples are presented: in the first, for allaandbthe four-genus is completely determined by the Tristram–Levine signature function; for the second, the recently defined Upsilon function of Ozsváth–Stipsicz–Szabó determines the four-genus for allaandb; for the third, a surprising interplay between signatures and Upsilon appears.

Connected Sums

1997 ◽  
pp. 159-177
Author(s):  
A. T. Fomenko ◽  
S. V. Matveev
Keyword(s):  
Connected Sums
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