Instanton Floer homology for lens spaces

TL;DR

This paper develops instanton Floer homology specifically for lens spaces and applies it to prove a non-decomposition result for a particular 4-manifold, revealing new insights into 4-manifold topology.

Contribution

It introduces a construction of instanton Floer homology for lens spaces and applies it to a novel non-decomposition theorem for certain 4-manifolds.

Findings

Constructed instanton Floer homology for lens spaces

Proved that a specific 4-manifold cannot be decomposed into two non-spin 4-manifolds with boundary lens spaces

Identified conditions on prime numbers related to the non-decomposition

Abstract

We construct instanton Floer homology for lens spaces $L(p,q)$. As an application, we prove that $X = \CP^2 # \CP^2$ does not admit a decomposition $X = X_1 \cup X_2$. Here $X_1$ and $X_2$ are oriented, simply connected, non-spin 4-manifolds with $b^+ = 1$ and with boundary $L(p, 2)$, and $p$ is a prime number of the form $16N+1$.