Obstructions to branch-decomposition of matroids

A ( δ , γ ) -net in a matroid M is a pair ( N , P ) where N is a minor of M, P is a set of series classes in N, | P | ⩾ δ , and the pairwise connectivity, in M, between any two members of P is at least γ. We prove that, for any finite field F , nets provide a qualitative characterization for branch-width in the class of F -representable matroids. That is, for an F -representable matroid M, we prove that: (1) if M contains a ( δ , γ ) -net where δ and γ are both very large, then M has large branch-width, and, conversely, (2) if the branch-width of M is very large, then M or M ∗ contains a ( δ , γ ) -net where δ and γ are both large.