Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLefschetz fibrations, intersection numbers, and representations of the framed braid group
430
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We examine the action of the fundamental group $Gamma$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $widetilde{mathcal B}$ on $m$ strings on a suitable space of $mtimes m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${mathcal S}_c:widetilde{mathcal B}to GL_m({mathbb{Z}}[Gamma])$ parametrized by distinguished configurations $c$ of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information.
rate research
Read More
We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.
We obtain infinitely many (non-conjugate) representations of 3-manifold fundamental groups into a lattice in the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of the projective plane along an arrangement of hyperplanes constructed by Hirzebruch. The 3-manifolds are related to a Lefschetz fibration of the complex hyperbolic orbifold.
We investigate two categorified braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer homology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group.
Let $G$ be a finitely generated group that can be written as an extension [ 1 longrightarrow K stackrel{i}{longrightarrow} G stackrel{f}{longrightarrow} Gamma longrightarrow 1 ] where $K$ is a finitely generated group. By a study of the BNS invariants we prove that if $b_1(G) > b_1(Gamma) > 0$, then $G$ algebraically fibers, i.e. admits an epimorphism to $Bbb{Z}$ with finitely generated kernel. An interesting case of this occurrence is when $G$ is the fundamental group of a surface bundle over a surface $F hookrightarrow X rightarrow B$ with Albanese dimension $a(X) = 2$. As an application, we show that if $X$ has virtual Albanese dimension $va(X) = 2$ and base and fiber have genus greater that $1$, $G$ is noncoherent. This answers for a broad class of bundles a question of J. Hillman.
In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
