On the geography of simply connected nonspin symplectic 4-manifolds with nonnegative signature

Anar Akhmedov; Sümeyra Sakalli

doi:10.1016/j.topol.2016.03.026

On the geography of simply connected nonspin symplectic 4-manifolds with nonnegative signature

Anar Akhmedov, Sümeyra Sakalli

School of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

In [8,5], the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP²#(2n-1)CP² for each integer n≥ 25, and the families of simply connected irreducible nonspin symplectic 4-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in [8,5]. In particular, we construct (i) an infinitely many irreducible symplectic and nonsymplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP²#(2n-1)CP² for each integer n≥ 12, and (ii) the families of simply connected irreducible nonspin symplectic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with positive signature and with more than one smooth structure. Our construction uses the complex surfaces of Hirzebruch and Bauer-Catanese on Bogomolov-Miyaoka-Yau line with c₁²=9χ_h=45, along with the exotic symplectic 4-manifolds constructed in [2,6,4,7,11].

Original language	English (US)
Pages (from-to)	24-45
Number of pages	22
Journal	Topology and its Applications
Volume	206
DOIs	https://doi.org/10.1016/j.topol.2016.03.026
State	Published - Jun 15 2016

Bibliographical note

Funding Information:
The first author would like to thank Professor Fabrizio Catanese for a useful email exchange on surfaces in [14] . The authors would like to thank the anonymous referee for the valuable comments on the previous version of this manuscript. A. Akhmedov was partially supported by NSF grants DMS-1065955 , DMS-1005741 and Sloan Research Fellowship . S. Sakallı was partially supported by NSF grants DMS-1065955 .

Publisher Copyright:
© 2016 Elsevier B.V.

Keywords

4-Manifolds
Exotic 4-manifolds
Geography of symplectic 4-manifolds

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10.1016/j.topol.2016.03.026

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title = "On the geography of simply connected nonspin symplectic 4-manifolds with nonnegative signature",

abstract = "In [8,5], the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP2#(2n-1)CP2 for each integer n≥ 25, and the families of simply connected irreducible nonspin symplectic 4-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in [8,5]. In particular, we construct (i) an infinitely many irreducible symplectic and nonsymplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP2#(2n-1)CP2 for each integer n≥ 12, and (ii) the families of simply connected irreducible nonspin symplectic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with positive signature and with more than one smooth structure. Our construction uses the complex surfaces of Hirzebruch and Bauer-Catanese on Bogomolov-Miyaoka-Yau line with c12=9χh=45, along with the exotic symplectic 4-manifolds constructed in [2,6,4,7,11].",

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author = "Anar Akhmedov and S{\"u}meyra Sakalli",

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N1 - Funding Information: The first author would like to thank Professor Fabrizio Catanese for a useful email exchange on surfaces in [14] . The authors would like to thank the anonymous referee for the valuable comments on the previous version of this manuscript. A. Akhmedov was partially supported by NSF grants DMS-1065955 , DMS-1005741 and Sloan Research Fellowship . S. Sakallı was partially supported by NSF grants DMS-1065955 . Publisher Copyright: © 2016 Elsevier B.V.

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N2 - In [8,5], the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP2#(2n-1)CP2 for each integer n≥ 25, and the families of simply connected irreducible nonspin symplectic 4-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in [8,5]. In particular, we construct (i) an infinitely many irreducible symplectic and nonsymplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP2#(2n-1)CP2 for each integer n≥ 12, and (ii) the families of simply connected irreducible nonspin symplectic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with positive signature and with more than one smooth structure. Our construction uses the complex surfaces of Hirzebruch and Bauer-Catanese on Bogomolov-Miyaoka-Yau line with c12=9χh=45, along with the exotic symplectic 4-manifolds constructed in [2,6,4,7,11].

AB - In [8,5], the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP2#(2n-1)CP2 for each integer n≥ 25, and the families of simply connected irreducible nonspin symplectic 4-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in [8,5]. In particular, we construct (i) an infinitely many irreducible symplectic and nonsymplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP2#(2n-1)CP2 for each integer n≥ 12, and (ii) the families of simply connected irreducible nonspin symplectic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with positive signature and with more than one smooth structure. Our construction uses the complex surfaces of Hirzebruch and Bauer-Catanese on Bogomolov-Miyaoka-Yau line with c12=9χh=45, along with the exotic symplectic 4-manifolds constructed in [2,6,4,7,11].

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On the geography of simply connected nonspin symplectic 4-manifolds with nonnegative signature

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