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Register a new userGeometric realizations of the multiplihedron and its complexification
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We realize Stasheffs multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.
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Let $ text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$, and let $fin text{Mod}(S_g)$ be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on $S_g$ that realizes $f$ as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of $ text{Mod}(S_g)$. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation $Psi: text{Mod}(S_g) to text{Sp}(2g; mathbb{Z})$.
Let $ text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$. Given a finite subgroup $H leq text{Mod}(S_g)$, let $text{Fix}(H)$ denote the set of fixed points induced by the action of $H$ on the Teichm{u}ller space $text{Teich}(S_g)$. The Nielsen realization problem, which was answered in the affirmative by S. Kerckhoff, asks whether $text{Fix}(H) eq emptyset$, for any given $H$. In this paper, we give an explicit description of $text{Fix}(H)$, when $H$ is cyclic. As consequences of our main result, we provide alternative proofs for two well known results, namely a result of Harvey on $text{dim}(text{Fix}(H))$, and a result of Gilman that characterizes irreducible finite order actions. Finally, we derive a correlation between the orders of irreducible cyclic actions and the filling systems on surfaces.
We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in terms of geometric realizations.
In this paper, we give geometric realizations of Lusztigs symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztigs symmetries.
We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant *-scalar curvature which satisfies the Kaehler condition at the point in question.
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