Bivariate topological complexity: a framework for coordinated motion planning

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 Pave\v{s}i\'c'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.