New exotic 4-manifolds via Luttinger surgery on Lefschetz fibrations

Anar Akhmedov, Kadriye Nur Saglam

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1 Scopus citations

Abstract

In [Small exotic 4-manifolds, Algebr. Geom. Topol. 8 (2008) 1781-1794], the first author constructed the first known example of exotic minimal symplectic CP2#5CP2and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to 3CP2#7CP2. The construction in [Small exotic 4-manifolds, Algebr. Geom. Topol. 8 (2008) 1781-1794] uses Yukio Matsumoto's genus two Lefschetz fibrations on T2×S2#4CP 2 over S2along with the fake symplectic S2×S2construction given in [Construction of symplectic cohomology S2×S2, Proc. Gökova Geom. Topol. Conf. 14 (2007) 36-48]. The main goal in this paper is to generalize the construction in [Small exotic 4-manifolds, Algebr. Geom. Topol. 8 (2008) 1781-1794] using the higher genus versions of Matsumoto's fibration constructed by Mustafa Korkmaz and Yusuf Gurtas on σk × S2#4nCP 2 for any k ≥ 2 and n = 1, and k ≥ 1 and n ≥ 2, respectively. Using our symplectic building blocks, we also construct new symplectic 4-manifolds with the free group of rank s ≥ 1, the free product of the finite cyclic groups, and various other finitely generated groups as the fundamental group.

Original languageEnglish (US)
Article number1550010
JournalInternational Journal of Mathematics
Volume26
Issue number1
DOIs
StatePublished - Jan 25 2015

Bibliographical note

Funding Information:
The authors are grateful to the referee for valuable comments and suggestions. A. Akhmedov was partially supported by NSF grants DMS-1065955, DMS-1005741 and Sloan Research Fellowship. N. Saglam was partially supported by NSF grant DMS-1065955.

Keywords

  • 4-manifolds
  • Lefschetz fibration
  • Luttinger surgery
  • exotic smooth structures
  • symplectic 4-manifolds
  • symplectic sum

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