Sympletic normal connect sum

McCarthy, John; Wolfson, Jon

doi:10.1016/0040-9383(94)90006-x

Cited by 73 publications

(62 citation statements)

References 11 publications

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“…In dimension 4 the situation is different [P1, P2, P3]. Using the Gompf symplectic sum construction [G1,MW], it is possible to construct families of closed 4-manifolds which are homeomorphic to (2m + 1)CP 2 #nCP 2 but not mutually diffeomorphic. It follows that (2m + 1)CP 2 #nCP 2 admits infinitely many symplectic structures which are all "exotic", since this manifold does not admit symplectic structures coming from the standard smooth structure, by the Taubes theorem.…”

Section: Examples Of Exotic Structures On Torimentioning

confidence: 99%

Exotic smooth structures and symplectic forms on closed manifolds

Hajduk¹,

Tralle

2007

Geom Dedicata

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Abstract. In this paper we discuss relations between symplectic forms and smooth structures on closed manifolds. Our main motivation is the problem if there exist symplectic structures on exotic tori. This is a symplectic generalization of a problem posed by Benson and Gordon. We give a short proof of the (known) positive answer to the original question of Benson and Gordon that there are no Kähler structures on exotic tori. We survey also other related results which give an evidence for the conjecture that there are no symplectic structures on exotic tori.

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Section: Examples Of Exotic Structures On Torimentioning

confidence: 99%

Exotic smooth structures and symplectic forms on closed manifolds

Hajduk¹,

Tralle

2007

Geom Dedicata

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“…[37]) implies that Z is a symplectic manifold, and its characteristic numbers are given by s (Z) = 3 and Xh(Z) = I. A result of Usher using the criteria of Hamilton-Kotschick [3 1, Corollary I], we proceed to compute its fundamental group using the Seifert-van Kampen theorem .…”

Section: Rafa El Torresmentioning

confidence: 99%

“…The geography problem for symplectic 4-manifolds with a given fundamental group [25,37] asks which homeomorphism classes arc realized by an irreducible sympleclic 4-manifold . The botany problem [26] asks how many diffeomorphism classes exist within a given homeomorphism type.…”

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confidence: 99%

Geography and Botany of Irreducible Non-Spin Symplectic 4-Manifolds With Abelian Fundamental Group

Torres

2013

Glasgow Math. J.

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Abstract. The geography and botany problems of irreducible non-spin symplectic 4-rnanifolds with a choice of fundamental group from {lp . "llp EB "ll.q, l, Z EB Zp, ZEB Z} are studied by building upon the recent progress obtained on the simply con nected realm. R esults on the botany of simply connected 4-manifolds not available in the literature are extended .2010 Mathematics Subject Classification. Primary 57R 17; Secondary 57M05, 54D051. Introduction. Topologists' understanding of smooth 4-manifolds has witnessed a drastic improvement during the last 20 years. Advances in symplectic geometry [8,25,31,35,[65][66][67]71] and inventions of gluing formulas [39,46] for diff eomorphism invariants (73] have paired elegantly with topological constructions [16,18,[19][20][21], offering a noteworthy insight into the smooth four-dimensional story. The most recent series of successes [2-4, 6, 11, 16, 21 ] has answered as many questions as it has raised new ones, and the 4-manifolds theory remains to be an intriguing and active area of research.Two factors responsible for the recent progress on the simply connected realm are the increase in the repertoire of techniques that manufacture symplectic 4-manifolds with sm all topological invariants, and a new perspective on the usage of already existing mechanisms of construction. The idea of using symplectic sums (25) of non-simply connected building blocks along genus 2 surfaces to kill fundamenta l groups in an efficient way was introduced in (1). Its immediate outcome was the

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“…The geography problem for symplectic 4-manifolds was originally posed by McCarthy and Wolfson in [15], and the first systematic study was carried out by Gompf in [12]. In this paper we will be concerned with the geography problem for closed simply connected spin symplectic 4-manifolds.…”

Section: Introductionmentioning

confidence: 99%

Geography of simply connected spin symplectic 4-manifolds

Akhmedov

Park

2010

Mathematical Research Letters

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Abstract. We present an algorithm that produces new families of closed simply connected spin symplectic 4-manifolds with nonnegative signature that are interesting with respect to the symplectic geography problem. In particular, for each odd integer q satisfying q ≥ 275, we construct infinitely many pairwise nondiffeomorphic irreducible smooth structures on the topological 4-manifold q(S 2 × S 2 ), the connected sum of q copies of S 2 × S 2 .

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Sympletic normal connect sum

Cited by 73 publications

References 11 publications

Exotic smooth structures and symplectic forms on closed manifolds

Exotic smooth structures and symplectic forms on closed manifolds

Geography and Botany of Irreducible Non-Spin Symplectic 4-Manifolds With Abelian Fundamental Group

Geography of simply connected spin symplectic 4-manifolds

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