scholarly journals Connected sums of knots and weakly reducible Heegaard splittings

2004 ◽  
Vol 141 (1-3) ◽  
pp. 1-20 ◽  
Author(s):  
Yoav Moriah
Keyword(s):  
Heegaard Splittings ◽  
Connected Sums ◽  
Weakly Reducible

scholarly journals Haken spheres for genus two Heegaard splittings

2017 ◽  
Vol 165 (3) ◽  
pp. 563-572 ◽  
Author(s):  
SANGBUM CHO ◽  
YUYA KODA
Keyword(s):  
Heegaard Splitting ◽  
Combinatorial Structure ◽  
Lens Spaces ◽  
Heegaard Splittings ◽  
Precise Description ◽  
Connected Sum ◽  
Connected Sums ◽  
Genus 2 ◽  
Genus Two

AbstractA manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S2 × S1, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the 3-sphere, S2 × S1 or a connected sum whose summands are lens spaces or S2 × S1, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.

CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

2004 ◽  
Vol 13 (06) ◽  
pp. 829-843 ◽  
Author(s):  
YOAV MORIAH ◽  
ERIC SEDGWICK
Keyword(s):  
Boundary Component ◽  
Closed Surface ◽  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Manifolds With Boundary ◽  
Minimal Genus ◽  
Essential Surfaces ◽  
Two Component ◽  
Single Boundary ◽  
Weakly Reducible

We show that there are infinitely many two component links in S3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.

Reducible handle additions to weakly reducible Heegaard splittings

2018 ◽  
Vol 242 ◽  
pp. 33-42
Author(s):  
Liang Liang ◽  
Fengling Li ◽  
Jingyan Li
Keyword(s):  
Heegaard Splittings ◽  
Weakly Reducible

Quasipositive Links and Connected Sums

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

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.

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