A stable classification of Lefschetz fibrations

We study the classification of Lefschetz fibrations up to stabilization by fiber sum operations. We show that for each genus there is a `universal' fibration f^0_g with the property that, if two Lefschetz fibrations over S^2 have the same Euler-Poincare characteristic and signature, the same numbers of reducible singular fibers of each type, and admit sections with the same self-intersection, then after repeatedly fiber summing with f^0_g they become isomorphic. As a consequence, any two compact integral symplectic 4-manifolds with the same values of (c_1^2, c_2, c_1.[w], [w]^2) become symplectomorphic after blowups and symplectic sums with f^0_g.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper6.abs.htm

Brick polytopes, lattice quotients, and Hopf algebras

This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic $k$-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic $k$-triangulations. We show that the fibers of this surjection are the classes of the congruence $\equiv^k$ on $\mathfrak{S}_n$ defined as the transitive closure of the rewriting rule $U ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W$ for letters $a < b_1, \dots, b_k < c$ and words $U, V_1, \dots, V_k, W$ on $[n]$. We then show that the increasing flip order on $k$-triangulations is the lattice quotient of the weak order by this congruence. Moreover, we use this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations, indexed by acyclic $k$-triangulations, and to describe the product and coproduct in this algebra and its dual in term of combinatorial operations on acyclic $k$-triangulations. Finally, we extend our results in three directions, describing a Cambrian, a tuple, and a Schr\"oder version of these constructions.Comment: 59 pages, 32 figure