• Title/Summary/Keyword: 4-manifolds

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REMARKS ON HIGHER TYPE ADJUNCTION INEQUALITIES OF 4-MANIFOLDS OF NON-SIMPLE TYPE

  • Kim, Jin-Hong
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.431-440
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    • 2002
  • Recently P. Ozsv$\'{a}$th Z. Szab$\'{o}$ proved higher type adjunction inequalities for embedded surfaces in 4-manifolds of non-simple type. The aim of this short paper is to give a simple and direct proof of such higher type adjunction inequalities for smoothly embedded surfaces with negative self-intersection number in smooth 4-manifolds of non-simple type. This will be achieved through a relation between the Seiberg-Witten invariants used to get adjunction inequalities of 4-manifolds of simple type and a blow-up formula.

SYMPLECTIC 4-MANIFOLDS VIA SYMPLECTIC SURGERY ON COMPLEX SURFACE SINGULARITIES

  • PARK, HEESANG;STIPSICZ, ANDRAS I.
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.4
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    • pp.1213-1223
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    • 2015
  • We examine a family of isolated complex surface singularities whose exceptional curves consist of two complex curves with high genera intersecting transversally. Topological data of smoothings of these singularities are determined. We use these computations to construct symplectic 4-manifolds by replacing neighborhoods of the exceptional curves with smoothings of the singularities.

AN EMBEDDED 2-SPHERE IN IRREDUCIBLE 4-MANIFOLDS

  • Park, Jong-Il
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.4
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    • pp.683-691
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    • 1999
  • It has long been a question which homology class is represented by an embedded 2-sphere in a smooth 4-manifold. In this article we study the adjunction inequality, one of main results of Seiberg-Witten theory in smooth 4-manifolds, for an embedded 2-sphere. As a result, we give a criterion which homology class cannot be represented by an embedded 2-sphere in some cases.

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Monodromy Groups on Knot Surgery 4-manifolds

  • Yun, Ki-Heon
    • Kyungpook Mathematical Journal
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    • v.53 no.4
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    • pp.603-614
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    • 2013
  • In the article we show that nondieomorphic symplectic 4-manifolds which admit marked Lefschetz fibrations can share the same monodromy group. Explicitly we prove that, for each integer g > 0, every knot surgery 4-manifold in a family {$E(2)_K{\mid}K$ is a bered 2-bridge knot of genus g in $S^3$} admits a marked Lefschetz fibration structure which has the same monodromy group.

Smooth structures on symplectic 4-manifolds with finite fundamental groups

  • Cho, Yong-Seung
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.619-629
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    • 1996
  • In studying smooth 4-manifolds the Donaldson invariant has played a central role. In [D1] Donaldson showed that non-vanishing Donaldson invariant of a smooth closed oriented 4-manifold X gives rise to the indecomposability of X. For instance, the complex algebraic suface X cannot decompose to a connected sum $X_1 #X_2$ with both $b_2^+(X_i) > 0$.

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ON SPIN ALTERNATING GROUP ACTIONS ON SPIN 4-MANIFOLDS

  • Kiyono, Kazuhiko;Liu, Ximin
    • Journal of the Korean Mathematical Society
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    • v.43 no.6
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    • pp.1183-1197
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    • 2006
  • Let X be a smooth, closed, connected spin 4-manifold with $b_1(X)=0$ and signature ${\sigma}-(X)$. In this paper we use Seiberg-Witten theory to prove that if X admits a spin alternating $A_4$ action, then $b^+_2(X)$ ${\geq}$ |${\sigma}{(X)}$|/8+3 under some non-degeneracy conditions.

ON THE TOPOLOGY OF DIFFEOMORPHISMS OF SYMPLECTIC 4-MANIFOLDS

  • Kim, Jin-Hong
    • Journal of the Korean Mathematical Society
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    • v.47 no.4
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    • pp.675-689
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    • 2010
  • For a closed symplectic 4-manifold X, let $Diff_0$(X) be the group of diffeomorphisms of X smoothly isotopic to the identity, and let Symp(X) be the subgroup of $Diff_0$(X) consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers {$n_1$, $n_2$, $\ldots$, $n_k$} and any non-negative integer m, there exists a closed symplectic (or K$\ddot{a}$hler) 4-manifold X with $b_2^+$ (X) > m such that the homologies $H_i$ of the quotient space $Diff_0$(X)/Symp(X) over the rational coefficients are non-trivial for all odd degrees i = $2n_1$ - 1, $\ldots$, $2n_k$ - 1. The basic idea of this paper is to use the local invariants for symplectic 4-manifolds with contact boundary, which are extended from the invariants of Kronheimer for closed symplectic 4-manifolds, as well as the symplectic compactifications of Stein surfaces of Lisca and Mati$\acute{c}$.