Counting hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants

TL;DR

This paper establishes asymptotic bounds on the number of hyperbolic four-manifolds with vanishing Seiberg-Witten invariants, showing they grow similarly to all hyperbolic four-manifolds as volume increases.

Contribution

It provides the first asymptotic bounds on the count of hyperbolic 4-manifolds with specific invariants, extending previous enumeration results.

Findings

Number of such manifolds grows asymptotically as v^{cv}

Bounds match the growth rate of all hyperbolic 4-manifolds

Supports existence of many hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants

Abstract

Ian Agol and Francesco Lin proved the existence of hyperbolic four-manifolds with vanishing Seiberg-Witten invariants. We prove that the number of such manifolds of volume at most $v$ is asymptotically bounded by $v^{cv}$ considered up to commensurability, which has the same form as the lower bound and upper bound of the number of hyperbolic four-manifolds of volume at most $v$ proved by Tsachik Gelander and Arie Levit.