Dual submanifolds in rational homology spheres

TL;DR

This paper characterizes the possible dimensions and codimensions of dual submanifolds in simply connected rational homology spheres, providing a complete classification of such triples.

Contribution

It offers a comprehensive classification of integral triples describing dual submanifolds in rational homology spheres, resolving a fundamental geometric problem.

Findings

Complete characterization of admissible triples (n; m_+, m_-)

Identification of conditions for dual submanifold existence

Clarification of the topological structure of rational homology spheres

Abstract

Let $\Sigma$ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds $M_+, M_-$ in $\Sigma$ are called dual to each other if the complement $\Sigma - M_+$ strongly homotopy retracts onto $M_-$ or vice-versa. In this paper we will give a complete answer of which integral triples $(n; m_+, m_-)$ can appear, where $n=dim \Sigma -1$, $m_+={codim}M_+ -1$ and $m_-={codim}M_- -1$.