Exotic Stein fillings with arbitrary fundamental group

Akhmedov, Anar; Özbağcı, Burak

doi:10.1007/s10711-017-0289-y

Cited by 16 publications

(21 citation statements)

References 39 publications

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“…Many Lefschetz fibrations with explicit monodromies and non-complex total spaces have been constructed using the (twisted) fiber sum operation (see for instance [32], [30], [17], [22], [3] 2 , [2], [8]). They are non-holomorphic, however, do not have any (−1)-section since they are decomposable.…”

Section: Lefschetz Fibrations With Non-complex Total Spacementioning

confidence: 99%

Nonholomorphic Lefschetz fibrations with (−1)-sections

2019

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We construct two types of non-holomorphic Lefschetz fibrations over S 2 with (−1)-sections -hence, they are fiber sum indecomposable-by giving the corresponding positive relators. One type of the two does not satisfy the slope inequality (a necessary condition for a fibration to be holomorphic) and has a simply-connected total space, and the other has a total space that cannot admit any complex structure in the first place. These give an alternative existence proof for non-holomorphic Lefschetz pencils without Donaldson's theorem.

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Section: Lefschetz Fibrations With Non-complex Total Spacementioning

confidence: 99%

Nonholomorphic Lefschetz fibrations with (−1)-sections

2019

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“…• If (Y, y) is Cohen-Macaulay and if the hyperplane section (f −1 (0), y) has an isolated singularity, property (1) implies that (f −1 (0), y) is also Cohen-Macaulay. Property (2) implies then that it is normal.…”

Section: Sweeping Out the Cone With Hyperplane Sectionsmentioning

confidence: 99%

“…Ohta and Ono [34] showed that there exist Milnor fillable contact 3-manifolds which admit an infinite number of minimal strong symplectic fillings, pairwise not homotopy equivalent. Later, Akhmedov and Ozbagci [1] proved that there exist Milnor fillable contact 3-manifolds which admit even an infinite number of Stein fillings pairwise non-diffeomorphic, but homeomorphic. Moreover, by varying the contact 3-manifold, the fundamental groups of such fillings exhaust all finitely presented groups.…”

Section: Introductionmentioning

confidence: 99%

On the smoothings of non-normal isolated surface singularities

Popescu-Pampu¹

2015

jsing

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We show that isolated surface singularities which are non-normal may have Milnor fibers which are non-diffeomorphic to those of their normalizations. Therefore, non-normal isolated singularities enrich the collection of Stein fillings of links of normal isolated singularities. We conclude with a list of open questions related to this theme.Comment: 14 pages, 1 figure. Compared to the first version on ArXiv, I added Remark 5.10, containing information communicated to me by J\'anos Koll\'ar. Proceedings of Singularities in Geometry and Applications III, Edinburgh, Scotland, 2013. J.-P. Brasselet, P. Giblin, V. Goryunov ed

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“…The later problem has been studied for the standard family of hyperelliptic Lefschetz fibrations (with total spaces CP 2 #(4g + 5)CP 2 ) in [27,31,39], using the computations in the mapping class group, and such results are useful in constructing (exotic) Stein fillings [1,3].…”

Section: Introductionmentioning

confidence: 99%

Constructing Lefschetz fibrations via daisy substitutions

Akhmedov

Monden²

2016

Kyoto J. Math.

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Abstract. Let M(V ) = M(n, F q ) denote the algebra of n × n matrices over F q , and let M(V ) U denote the (maximal reducible) subalgebra that normalizes a given r-dimensional subspace U of V = F n q where 0 < r < n. We prove that the density of non-cyclic matrices in M(V ) U is at least q −2 1 + c 1 q −1 , and at most q −2 1 + c 2 q −1 , where c 1 and c 2 are constants independent of n, r, and q. The constants c 1 = − AMS Subject Classification (2010): 15B52, 60B20, 68W40 The main resultThe Meat-axe is an algorithm often used to test whether a given group or algebra of matrices over a finite field acts irreducibly on the underlying vector space, see [P,HR,NP2]. It uses random selection to find a 'good' matrix, and if successful is able to determine whether the action is reducible or irreducible. One definition of a 'good' matrix in this context is a cyclic matrix. (A matrix is cyclic if its characteristic and minimal polynomials are equal.) The density of cyclic matrices in absolutely irreducible groups and algebras is constrained by the following result of Neumann and the fourth author [NP1, Theorem 4.1]. The probability(1 − q −1 )(1 − q −2 ) for all d 2 and q 2., then P 1,q = 0 because each 1 × 1 matrix is cyclic. Bounds on the proportion of non-cyclic matrices in irreducible-butnot-absolutely-irreducible matrix algebras are also available in [NP1].This note shows that cyclic matrices are less dense in maximal reducible matrix algebras than full matrix algebras, with density 1 − c(q)q −2 rather than 1 − c ′ (q)q −3 where c(q), c ′ (q) are bounded functions. We do not know how to estimate the density δ of cyclic matrices in arbitrary non-maximal reducible algebras. Since 0 δ 1, our lower bound q −2 (1 + c 1 q −1 ) < δ is unhelpful if c 1 < −q for some choice of q. Similarly, our upper bound δ < q −2 (1 + c 2 q −1 ) is unhelpful if c 2 > q(q 2 − 1). We go to some effort to find helpful bounds for all values of q. While motivated by a complexity analysis of the Meat-axe algorithm, we feel that this problem has broader interest. A modification of Norton's Irreducibility Test, called the Cyclic Irreducibility Test, was presented in [NP2]. It was shown to be a Monte Carlo algorithm that proved irreducibility of a finite irreducible matrix algebra A provided a cyclic pair was found, that is a pair (v, X) where X is a cyclic matrix in A, and v is a cyclic vector for X. It was hoped that cyclic pairs in reducible matrix algebras, if such exist, could be used to construct a proper A-invariant subspace. However, it was not known which reducible algebras A might contain a sufficiently high proportion of cyclic matrices to make this approach worth exploring. In this paper we prove that finite maximal reducible matrix algebras do indeed have a plentiful supply of cyclic elements, with the proportion slightly less than that for the full matrix algebra. A variant of the Cyclic Irreducibility Test is given in [B, p. 141].Notation A. The following notation will be used throughout the paper. F = F q a finite field with q eleme...

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Exotic Stein fillings with arbitrary fundamental group

Cited by 16 publications

References 39 publications

Nonholomorphic Lefschetz fibrations with (−1)-sections

Nonholomorphic Lefschetz fibrations with (−1)-sections

On the smoothings of non-normal isolated surface singularities

Constructing Lefschetz fibrations via daisy substitutions

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