Connected sums of unstabilized Heegaard splittings are unstabilized

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.

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Quasipositive Links and Connected Sums

Functional Analysis and Its Applications ◽

10.1134/s0016266320010098 ◽

2020 ◽

Vol 54 (1) ◽

pp. 64-67

Author(s):

S. Yu. Orevkov

Keyword(s):

Connected Sums

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Constructing symplectic manifolds

10.1093/oso/9780198794899.003.0008 ◽

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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THE RESOLVENT AND RIESZ TRANSFORM ON CONNECTED SUMS OF MANIFOLDS WITH DIFFERENT ASYMPTOTIC DIMENSIONS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972721000496 ◽

2021 ◽

pp. 1-2

Author(s):

DANIEL NIX

Keyword(s):

Riesz Transform ◽

Connected Sums

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Local detection of strongly irreducible Heegaard splittings

Topology and its Applications ◽

10.1016/s0166-8641(97)00184-3 ◽

1998 ◽

Vol 90 (1-3) ◽

pp. 135-147 ◽

Cited By ~ 28

Author(s):

Martin Scharlemann

Keyword(s):

Heegaard Splittings ◽

Strongly Irreducible ◽

Local Detection

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Local uniqueness of certain geodesics related to Heegaard splittings

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518420038 ◽

2018 ◽

Vol 27 (09) ◽

pp. 1842003

Author(s):

Liang Liang ◽

Fengling Li ◽

Fengchun Lei ◽

Jie Wu

Keyword(s):

Heegaard Splitting ◽

Sufficient Condition ◽

Heegaard Splittings ◽

Local Uniqueness ◽

Uniqueness Property

Suppose [Formula: see text] is a Heegaard splitting and [Formula: see text] is an essential separating disk in [Formula: see text] such that a component of [Formula: see text] is homeomorphic to [Formula: see text], [Formula: see text]. In this paper, we prove that if there is a locally complicated simplicial path in [Formula: see text] connecting [Formula: see text] to [Formula: see text], then the geodesic connecting [Formula: see text] to [Formula: see text] is unique. Moreover, we give a sufficient condition such that [Formula: see text] is keen and the geodesic between any pair of essential disks on the opposite sides has local uniqueness property.

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