# Volume 19 (2015) No. 1

## Volume 19 (2015) No. 1

### Real projective space as a space of planar polygons

Donald M. Davis

**Abstract:**

We describe an explicit homeomorphism between real projective space $RP^{n-3}$ and the space $\overline{M}_{n,n-2}$ of all isometry classes of $n$- gons in the plane with one side of length $n-2$ and all other sides of length 1. This makes the topological complexity of real projective space more relevant to robotics.

### Motion planning in tori revisited

Jesús González, Bárbara Gutiérrez, Aldo Guzmán,Cristhian Hidber, María Mendoza, Christopher Roque

**Abstract:**

The topological complexity (TC) of the complement of a complex hyperplane arrangement, which is either linear generic or else affine in general position, has been computed by Yuzvinsky. This is accomplished by noticing that efficient homotopy models for such spaces are given by skeletons of Cartesian powers of circles. Soon after, Cohen and Pruidze noticed that the topological complexity of the complement of the corresponding redundant subspace arrangement, as well as of right-angled Artin groups, can be obtained by considering general subcomplexes of cartesian powers of higher dimensional spheres. Unfortunately Cohen-Pruidze’s TC-calculations are flawed, and our work describes and mends the problems in order to validate the extended applications. In addition, we generalize Farber-Cohen’s computation of the topological complexity of oriented surfaces, now to the realm of Rudyak’s higher topological complexity.

### Toeplitz operators with piecewise quasicontinuous symbols

Breitner Ocampo

**Abstract:**

For a fixed subset of the unit circle $\partial\mathbb{D}$, $\Lambda:=\{ \lambda_1, \lambda_2, \dots, \lambda_n\}$, we define the algebra $PC$ of piecewise continuous functions in $\partial\mathbb{D} \setminus \Lambda$ with one sided limits at each point $\lambda_k \in \Lambda$. Besides, we let $QC$ stands for the $C^*$-algebra of quasicontinuous functions on $\partial \mathbb{D}$ defined by D. Sarason in [5]. We define then $PQC$ as the $C^*$-algebra generated by $PC$ and $QC$.

$\mathcal{A}^2(\mathbb{D})$ stands for the Bergman space of the unit disk $\mathbb{D}$, that is, the space of square integrable and analytic functions defined on $\mathbb{D}$. Our goal is to describe $\mathcal{T}_{PQC}$, the algebra generated by Toeplitz operators whose symbols are certain extensions of functions in $PQC$ acting on $\mathcal{A}^2(\mathbb{D})$. Of course, a function defined on $\partial\mathbb{D}$ can be extended to the disk in many ways. The more natural extensions are the harmonic and the radial ones. In the paper we describe the algebra $\mathcal{T}_{PQC}$ and we prove that this description does not depend on the extension chosen.