Lattice points, Dedekind-Rademacher sums and a conjecture of Kronheimer and Mrowka

TL;DR

This paper connects lattice point enumeration in simplices with Dedekind-Rademacher sums and proves a special case of a conjecture linking Seiberg-Witten-Floer homology and the Casson invariant for certain 3-manifolds.

Contribution

It introduces a novel method to express lattice points using Dedekind-Rademacher sums and verifies a conjecture for Brieskorn spheres with up to four singular fibers.

Findings

Expressed lattice points via Dedekind-Rademacher sums

Proved the conjecture for specific Brieskorn spheres

Linked Seiberg-Witten-Floer homology Euler characteristic to Casson invariant

Abstract

We express the number of lattice points inside certain simplices via Dedekind-Rademacher sums. As an application, we prove a conjecture of Kronheimer and Mrowka in the special case of Brieskorn spheres (with at most 4 singular fibers). This conjecture relates the Euler characteristic of the Seiberg-Witten-Floer homology to the Casson invariant.