Cochran's $\beta^i$ invariants via twisted Whitney towers

TL;DR

This paper demonstrates that Cochran's link invariants can be computed using special twisted Whitney towers in the 4-ball, revealing a new method for extracting multiple invariants simultaneously.

Contribution

It introduces Cochran towers, a new class of twisted Whitney towers, enabling the computation of multiple Cochran invariants at once, linking them to intersection invariants.

Findings

Cochran invariants are computable via intersection invariants of Cochran towers.

Cochran towers allow simultaneous extraction of multiple invariants.

The invariants are integer lifts of Milnor invariants.

Abstract

We show that Tim Cochran's invariants $\beta^i(L)$ 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 $\beta^i$ 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 $\beta^i(L)$ are integer lifts of certain Milnor invariants.