Constructing Lefschetz fibrations via daisy substitutions

Abstract

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...

Symplectic surgeries along certain singularities and new Lefschetz fibrations

“…The generalized star relation surgery. Let us now introduce the symplectic surgery operation, which corresponds to the generalized star relation (t a 0 t a 1 t a 2 · · · t a 2g+1 ) 2g+1 = t c 1 t c 2 g t c 3 . Let S g be the Stein 4-manifold determined by the Lefschetz fibration (t a 0 t a 1 t a 2 · · · t a 2g+1 ) 2g+1 in Γ 3 g .…”

“…Motivated by the above mentioned results and problems, our goal in this paper is to construct new 4-dimensional symplectic surgery operations arising from the generalized star relations (GSR for short) (t a 0 t a 1 t a 2 · · · t a 2g+1 ) 2g+1 = t b 1 t g b 2 t b 3 in the mapping class group Γ g, 3 of an orientable surface of genus g ≥ 1 with 3 boundary components. By applying the sequence of GSR substitutions, chain substitutions, and conjugations to the families of hyperelliptic words (c 1 c 2 · · · c 2g−1 c 2g c 2g+1 2 c 2g c 2g−1 · · · c 2 c 1 ) 2n = 1, (c 1 c 2 · · · c 2g c 2g+1 ) (2g+2)n = 1, and (c 1 c 2 · · · c 2g−1 c 2g ) 2(2g+1)n = 1 in the mapping class group of the closed orientable surface of genus g for any g ≥ 1, we also construct families of Lefschetz fibrations.…”

Deformation of singular fibers of genus two fibrations and small exotic symplectic 4-manifolds

“…In the case of the second relation, the total spaces of corresponding genus g Lefschetz fibrations over S 2 are also well-known families of complex surfaces. For example, Y (2) = K3#2CP 2 , and the Lefschetz fibration structure given arises from a well-known pencil in the K3 surface with two base points (see for example references [17,9]).…”

“…and the Lefschetz fibration structure given arises from a well-known pencil in the K3 surface with two base points (see for example references [17,9]).…”

The existence of an indecomposable minimal genus two Lefschetz fibration

“…Therefore, we suppose that any genus-2 Lefschetz fibrations of type (14,13) are a fiber sum of Lefschetz fibrations of types (n 1 , s 1 ) and (n 2 , s 2 ). Since then, by Lemma 2, the following pairs (n, s) are not realizable: (1,5), (2,4), (8,1), (10, 0), • If n + 2s = 20, then (n, s) = (0, 10), (2,9), (4, 8),…”