Cochran's $β^i$ invariants via twisted Whitney towers

Abstract

We show that Tim Cochran's invariants $β^i(L)$ of a $2$-component link $L$ in the $3$--sphere can be computed as intersection invariants of certain 2-complexes in the $4$--ball with boundary $L$. These 2-complexes are special types of twisted Whitney towers, which we call {\em Cochran towers}, and which exhibit a new phenomenon: A Cochran tower of order $2k$ allows the computation of the $β^i$ invariants for all $i\leq k$, i.e. simultaneous extraction of invariants from a Whitney tower at multiple orders. This is in contrast with the order $n$ Milnor invariants (requiring order $n$ Whitney towers) and consistent with Cochran's result that the $β^i(L)$ are integer lifts of certain Milnor invariants.