Proper topological complexity

Proper topological complexity

Hybrid | Sala de seminários | online

2023-07-14 - 15:00

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

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