Monodromy substitutions and rational blowdowns

Endo, Hidenori; Mark, Thomas E.; Horn-Morris, Jeremy Van

doi:10.1112/jtopol/jtq041

Cited by 40 publications

(117 citation statements)

References 13 publications

Supporting

Mentioning

116

Contrasting

Order By: Relevance

“…Recall that the Floer homology group y HF pLpq, rq, sq in any spin c structure s is 1-dimensional over F, and lies in a grading denoted dpLpq, rq, sq P Q. These grading levels were computed recursively by Ozsváth and Szabó in [27,Proposition 4.8]; in our orientation convention their formula reads (10) dpLpq, rq, iq " 1 4qr`q r´p2i`1´q´rq 2˘´d pLpr, qq, iq.…”

Section: 2mentioning

confidence: 99%

Naturality of Heegaard Floer invariants under positive rational contact surgery

Mark

Tosun

2018

J. Differential Geom.

Self Cite

View full text Add to dashboard Cite

For a nullhomologous Legendrian knot in a closed contact 3manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery, and we characterize the spin c structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery on a Legendrian knot in the standard 3sphere, generalizing previous results of Lisca-Stipsicz and Golla. In fact our methods allow direct calculation of the contact invariant in terms of the rational surgery mapping cone of Ozsváth and Szabó. The proof involves a construction called reducible open book surgery, which reduces in special cases to the capping-off construction studied by Baldwin.

show abstract

Section: 2mentioning

confidence: 99%

Naturality of Heegaard Floer invariants under positive rational contact surgery

Mark

Tosun

2018

J. Differential Geom.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Theorem 17 ([14], [12] (p = 2), [13] (p ≥ 3)). Let ̺, ̺ ′ be positive relators of Γ g , and let X ̺ , X ̺ ′ be the corresponding Lefschetz fibrations over S 2 , respectively.…”

Section: Rational Blowdownmentioning

confidence: 99%

“…Recently there has been much interest in trying to understand the topological interpretation of various relations in the mapping class group. A particularly well understood case is the daisy relation, which corresponds to the symplectic operation of rational blowdown [12,13]. Another interesting problem, which is still open, is whether any Lefschetz fibration over S 2 admits a section (see for example [36]).…”

Section: Introductionmentioning

confidence: 99%

Constructing Lefschetz fibrations via daisy substitutions

Akhmedov

Monden²

2016

Kyoto J. Math.

View full text Add to dashboard Cite

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...

show abstract

“…The referees have suggested that Endo's idea may be extended in an inductive manner to give similar examples with base and fibre of genus g, for any g ≥ 3. We should use the "daisy chain relation" of [9], which is also the "generalized lantern relation" of [2], and is an iteration of the lantern relation.…”

Section: Bundles With Hyperbolic Fibrementioning

confidence: 99%

Sections of surface bundles

Hillman

2015

Geometry &Amp; Topology Monographs

View full text Add to dashboard Cite

Let pW E ! B be a bundle projection with base B and fibre F aspherical closed connected surfaces. We review what algebraic topology can tell us about such bundles and their total spaces and then consider criteria for p to have a section. In particular, we simplify the cohomological obstruction, and show that the transgression d 2 2;0 in the homology LHS spectral sequence of a central extension is evaluation of the extension class. We also give several examples of bundles without sections.

show abstract

Monodromy substitutions and rational blowdowns

Cited by 40 publications

References 13 publications

Naturality of Heegaard Floer invariants under positive rational contact surgery

Naturality of Heegaard Floer invariants under positive rational contact surgery

Constructing Lefschetz fibrations via daisy substitutions

Sections of surface bundles

Contact Info

Product

Resources

About