Bivariate topological complexity: a framework for coordinated motion planning
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arXiv
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| Format: | Preprint |
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2026
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| author | Calcines, Jose Manuel Garcia Alarcon, Jose Antonio Vilches |
| author_facet | Calcines, Jose Manuel Garcia Alarcon, Jose Antonio Vilches |
| contents | We introduce a bivariate version of topological complexity, $\mathrm{TC}(f,g)$, associated with two continuous maps $f\colon X\to Z$ and $g\colon Y\to Z$. This invariant measures the minimal number of continuous motion planning rules required to coordinate trajectories in $X$ and $Y$ through a shared target space $Z$. It recovers Farber's classical topological complexity when $f=g=\mathrm{id}_X$ and Pavešić's map-based invariant when one of the maps is the identity.
We develop a structural theory for $\mathrm{TC}(f,g)$, including symmetry, product inequalities, stability properties, and a collaboration principle showing that, when one of the maps is a fibration, the complexity of synchronization is controlled by the other. We also introduce a homotopy-invariant bivariate complexity $\mathrm{TC}_H(f,g)$ of Scott type, defined via homotopic distance, and study its relationship with the strict invariant.
Concrete examples reveal rigidity phenomena with no analogue in the classical case, including strict gaps between $\mathrm{TC}(f,g)$ and $\mathrm{TC}_H(f,g)$ and situations where synchronization becomes impossible. Cohomological estimates provide computable obstructions in both the strict and homotopy-invariant settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16059 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bivariate topological complexity: a framework for coordinated motion planning Calcines, Jose Manuel Garcia Alarcon, Jose Antonio Vilches Algebraic Topology 55M30 (Primary) 55P99, 55S40 (Secondary) We introduce a bivariate version of topological complexity, $\mathrm{TC}(f,g)$, associated with two continuous maps $f\colon X\to Z$ and $g\colon Y\to Z$. This invariant measures the minimal number of continuous motion planning rules required to coordinate trajectories in $X$ and $Y$ through a shared target space $Z$. It recovers Farber's classical topological complexity when $f=g=\mathrm{id}_X$ and Pavešić's map-based invariant when one of the maps is the identity. We develop a structural theory for $\mathrm{TC}(f,g)$, including symmetry, product inequalities, stability properties, and a collaboration principle showing that, when one of the maps is a fibration, the complexity of synchronization is controlled by the other. We also introduce a homotopy-invariant bivariate complexity $\mathrm{TC}_H(f,g)$ of Scott type, defined via homotopic distance, and study its relationship with the strict invariant. Concrete examples reveal rigidity phenomena with no analogue in the classical case, including strict gaps between $\mathrm{TC}(f,g)$ and $\mathrm{TC}_H(f,g)$ and situations where synchronization becomes impossible. Cohomological estimates provide computable obstructions in both the strict and homotopy-invariant settings. |
| title | Bivariate topological complexity: a framework for coordinated motion planning |
| topic | Algebraic Topology 55M30 (Primary) 55P99, 55S40 (Secondary) |
| url | https://arxiv.org/abs/2601.16059 |