**José Calcines**

*Univ. de la Laguna, Spain.*

The topological complexity of a space $X$ is defined as $\mbox{TC}(X) := \mbox{secat}(\pi)$, which corresponds to the sectional category (or __Schwarz__ genus) of the end-points evaluation __fibration__ $\pi: X^I \rightarrow X \times X$. Here, $\pi(\alpha) = (\alpha(0), \alpha(1))$. This numerical __homotopy__ invariant was introduced by __Farber__ to analyze the topological instabilities in robotics motion planning algorithms.Another interesting numerical __homotopy__ invariant is the __Lusternik__-__Schnirelmann__ category (or LS-category for short). It originally aimed to provide a lower bound, denoted as $\mbox{cat}(M)$, for the number of critical points in any smooth map $f: M \rightarrow \mathbb{R}$, where $M$ is a closed smooth manifold. However, conventional __homotopy__ __invariants__, including numerical ones like TC or cat, do not accurately capture the behavior and geometry of non-compact spaces "at infinity". Proper __homotopy__ theory was developed to address this limitation. Notably, proper LS category was successfully introduced and developed by R. Ayala, E. __Domínguez__, A. M\'arquez, and A. __Quintero__.In this work, our goal is to introduce a version of topological complexity in the proper setting, following the same approach as proper LS category. It is worth noting that the proper setting imposes constraints on available __homotopy__ constructions. For instance, products and __fibrations__ are not feasible in the proper setting. This limitation poses a significant obstacle in developing topological complexity within proper __homotopy__ theory. To overcome these challenges, we consider the category $\mathbf{E}$ of exterior spaces.

Date and Time: 14th July, __15h__

Venue: Sala de Seminários do CMAT, Edif. 6 - 3.08