Contact domination
Sekh Kiran Ajij, Ritwik Chakraborty, Balarka Sen
TL;DR
The paper proves that every closed connected oriented odd-dimensional manifold $M^{2n+1}$ is dominated by a tight Liouville-fillable contact manifold $Y^{2n+1}$ via a map of strictly positive degree, extending Fine–Panov's domination to the odd-dimensional setting. The authors construct $Y$ as a Liouville boundary and employ asymptotically contact-holomorphic divisors together with Donaldson-type divisors to enforce a positive-degree domination, while showing this dominating manifold can be chosen Liouville-fillable but not Weinstein-fillable in general. A key technical component is the relative divisor technology (IMP) and the interaction with Donaldson divisors to ensure the required homological positivity, together with a homological-intersection analysis. They also provide obstructions and positive results for Weinstein-fillable domination and prove that certain asymptotically holomorphic divisors need not be Weinstein-fillable when built from nontrivial line bundles, highlighting a sharp contrast with the trivial-line case.
Abstract
In this note, we prove that every closed connected oriented odd-dimensional manifold admits a map of non-zero degree (i.e., a domination) from a tight contact manifold of the same dimension. This provides an odd-dimensional counterpart of a symplectic domination result due to Joel Fine and Dmitri Panov. We prove that the dominating contact manifold can be ensured to be Liouville-fillable, but not Weinstein-fillable in general. We discuss an application for contact divisors arising as zero sets of asymptotically contact-holomorphic sections.