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Register a new userA Geometric Proof of a Faithful Linear-Categorical Surface Mapping Class Group Action
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Authors
Kyler Siegel
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We give completely combinatorial proofs of the main results of [3] using polygons. Namely, we prove that the mapping class group of a surface with boundary acts faithfully on a finitely-generated linear category. Along the way we prove some foundational results regarding the relevant objects from bordered Heegaard Floer homology,
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We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus $g$, and determine its abelianization for $gge4$.
Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uniformly perfect, and that the map from their second bounded cohomology to ordinary cohomology is not injective.
In this paper, we discuss relations among several invariants of 3-manifolds including Meyers function, the eta-invariant, the von Neumann rho-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.
We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.
For $ggeq 2$, let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give a theoretical algorithm for computing its roots up to conjugacy. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in $text{Mod}(S_g)$, the Torelli group, the level-$m$ subgroup of $text{Mod}(S_g)$, and the commutator subgroup of $text{Mod}(S_2)$. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class $F$ is $3q(F)(g+1)(g+2)$, realized by the roots of $T_c^{q(F)}$, where $c$ is a separating curve in $S_g$ of genus $[g/2]$ and $q(F)$ is a unique positive integer associated with the conjugacy class of $F$. Finally, for $ggeq 3$ we show that any pseudo-periodic having a nontrivial periodic component that is not the hyperelliptic involution, normally generates $text{Mod}(S_g)$. Consequently, we establish there always exist roots of bounding pair maps and powers of Dehn twists that normally generate $text{Mod}(S_g)$.
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