A symplectic manifold homeomorphic but not diffeomorphic toCP2#3CP2

Baldridge, Scott; Kirk, Paul

doi:10.2140/gt.2008.12.919

Cited by 24 publications

(70 citation statements)

References 19 publications

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“…Choices of base points are fundamental in the application of the Seifert-van Kampen theo rem, which is to be used with care. For these matters we build greatly in the analysis done in [11][12][13]. In particular, we have the following result.…”

Section: Background Resultsmentioning

confidence: 98%

“…2 #kCl? 2 (1,6,4,11,13,16] . In [21 ), Fintushel and Stem introduced a procedure to unvei l exotic smooth structures on these manifolds by performing surgeries on nullhomologous tori that are contained inside rational surfaces (cf.…”

Section: } Then X G Has the 00 2 -Propertymentioning

confidence: 99%

“…Its immediate outcome was the construction of an exotic symplectic CIP 2 #5CIP 2 and, later on, the existence of an exotic symplectic Clfl> 2 #3CIP 2 [4] was put on display. Shortly after, Luttinger surgery [8,35) was introduced to the list of symplectic constructions in [11,56]. The combinations of these techniques yielded another construction of an exotic symplectic Clfl> 2 #3CIP 2 in [11,16] .…”

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confidence: 99%

“…Shortly after, Luttinger surgery [8,35) was introduced to the list of symplectic constructions in [11,56]. The combinations of these techniques yielded another construction of an exotic symplectic Clfl> 2 #3CIP 2 in [11,16] . Eventually these techniques were successful in unveiling exotic smooth structures on CIP 2 # 2Clfl> 2 (6,21], the 4-manifolds with the smallest Euler characteristic known to admit exotic smooth structures at the time this paper was written.…”

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confidence: 99%

“…Carve out the loop a q and glue in a copy of S 2 x D 2 in order to kill the generator that corresponds to the £'.-factor in rr 1 • there exists a n m e N for which there is a n element X, 11 e {X 11 } that is symplectic, while Xm is non-symplectic for every m =fa 11; we will assume m = 1;…”

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confidence: 99%

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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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Section: Background Resultsmentioning

confidence: 98%

Section: } Then X G Has the 00 2 -Propertymentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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

Torres

2013

Glasgow Math. J.

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Genus-3 Lefschetz Fibrations and Exotic 4-Manifolds with $$b_{2}^{+}=3$$

Altunöz

2021

Mediterr. J. Math.

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Small Lefschetz fibrations and exotic 4-manifolds

Baykur

Korkmaz

2016

Math. Ann.

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Abstract. We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP 2 #p CP 2 for p = 7, 8, 9, and to 3CP 2 #q CP 2 for q = 12, . . . , 19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with b + ≤ 3, with one possible exception when b + = 3. Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smallest possible example, which follow from a reverse engineering procedure we introduce for this setting. We also derive exotic minimal symplectic 4-manifolds in the homeomorphism classes of CP 2 #4CP 2 and 3CP 2 #6CP 2 from small Lefschetz fibrations over surfaces of higher genera.

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A symplectic manifold homeomorphic but not diffeomorphic toCP²#3CP²

Abstract: In this paper we construct a minimal symplectic 4-manifold and prove it is homeomorphic but not diffeomorphic to CP 2 #3CP

Cited by 24 publications

References 19 publications

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

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

Genus-3 Lefschetz Fibrations and Exotic 4-Manifolds with $$b_{2}^{+}=3$$

Small Lefschetz fibrations and exotic 4-manifolds

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