Cap-prodcut structures on the Fintushel-Stern spectral sequence

TL;DR

This paper establishes a cap-product structure on the Fintushel-Stern spectral sequence, leading to a new topological property of instanton Floer homology that combines finite and infinite dimensional classes.

Contribution

It introduces a well-defined cap-product structure on the spectral sequence and the induced structure on instanton Floer homology, revealing novel topological interactions.

Findings

Cap-product structure on the spectral sequence is well-defined.

Induces a new cap-product structure on instanton Floer homology.

Reveals a topological property involving finite and infinite dimensional classes.

Abstract

We show that there is a well-defined cap-product structure on the Fintushel-Stern spectral sequence. Hence we obtain the induced cap-product structure on the ${\BZ}_8$-graded instanton Floer homology. The cap-product structure provides an essentially new property of the instanton Floer homology, from a topological point of view, which multiplies a finite dimensional cohomology class by an infinite dimensional homology class (Floer cycles) to get another infinite dimensional homology class.