Simplicial Maps of the Complexes of Curves on Nonorientable Surfaces

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\lambda$ be a simplicial map of the complex of curves, $\mathcal{C}(N)$, on $N$ which satisfies the following: $[a]$ and $[b]$ are connected by an edge in $\mathcal{C}(N)$ if and only if $\lambda([a])$ and $\lambda([b])$ are connected by an edge in $\mathcal{C}(N)$ for every pair of vertices $[a], [b]$ in $\mathcal{C}(N)$. We prove that $\lambda$ is induced by a homeomorphism of $N$ if $(g, n) \in \{(1, 0), (1, 1), (2, 0)$, $(2, 1), (3, 0)\}$ or $g + n \geq 5$. Our result implies that superinjective simplicial maps and automorphisms of $\mathcal{C}(N)$ are induced by homeomorphisms of $N$.Comment: 13 pages, 6 figures. The paper was shortened and reorganize

Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups

Let $S$ be a closed, connected, orientable surface of genus at least 3, $\mathcal{C}(S)$ be the complex of curves on $S$ and $Mod_S^*$ be the extended mapping class group of $S$. We prove that a simplicial map, $\lambda: \mathcal{C}(S) \to \mathcal{C}(S)$, preserves nondisjointness (i.e. if $\alpha$ and $\beta$ are two vertices in $\mathcal{C}(S)$ and $i(\alpha, \beta) \neq 0$, then $i(\lambda(\alpha), \lambda(\beta)) \neq 0$) iff it is induced by a homeomorphism of $S$. As a corollary, we prove that if $K$ is a finite index subgroup of $Mod_S^*$ and $f: K \to Mod_S^*$ is an injective homomorphism, then $f$ is induced by a homeomorphism of $S$ and $f$ has a unique extension to an automorphism of $Mod_S^*$.Comment: 34 pages, 17 figure

On the arc and curve complex of a surface

We study the {\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of $AC(S)$ coincides with the natural image of the extended mapping class group of $S$ in that group. We also show that for any vertex of $AC(S)$, the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in $S$ that represents that vertex. We also give a proof of the fact if $S$ is not a sphere with at most three punctures, then the natural embedding of the curve complex of $S$ in $AC(S)$ is a quasi-isometry. The last result, at least under some slightly more restrictive conditions on $S$, was already known. As a corollary, $AC(S)$ is Gromov-hyperbolic.Comment: Added references, added some results about special surfaces and corrected some misprint

A Deep Incremental Boltzmann Machine for Modeling Context in Robots

Context is an essential capability for robots that are to be as adaptive as possible in challenging environments. Although there are many context modeling efforts, they assume a fixed structure and number of contexts. In this paper, we propose an incremental deep model that extends Restricted Boltzmann Machines. Our model gets one scene at a time, and gradually extends the contextual model when necessary, either by adding a new context or a new context layer to form a hierarchy. We show on a scene classification benchmark that our method converges to a good estimate of the contexts of the scenes, and performs better or on-par on several tasks compared to other incremental models or non-incremental models.Comment: 6 pages, 5 figures, International Conference on Robotics and Automation (ICRA 2018