Inserted: 2 oct 2014
Last Updated: 2 oct 2014
We show that for a metric space with an even number of points there is a $1$-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the duality gives a version of global calibrations for $1$-chains with coefficients in $\mathbb Z_2$. Finally we extend the results to infinite metric spaces and present a notion of ``matching dimension'' which arises naturally.