Rabinowitz Floer homology as a Tate vector space

TL;DR

This paper establishes Rabinowitz Floer homology as a Tate vector space within the framework of linearly topologized vector spaces, enabling a topological interpretation of Poincaré duality and algebraic structures.

Contribution

It introduces the framework of linearly topologized vector spaces for Floer homologies, proving Rabinowitz Floer homology is a Tate vector space and developing related duality and tensor product theories.

Findings

Rabinowitz Floer homology is a locally linearly compact (Tate) vector space.

Poincaré duality holds in the topological sense for Rabinowitz Floer homology.

The graded Frobenius algebra structure is compatible with the topological framework.

Abstract

We show that the category of linearly topologized vector spaces over discrete fields constitutes the correct framework for algebraic structures on Floer homologies with field coefficients. Our case in point is the Poincar\'e duality theorem for Rabinowitz Floer homology. We prove that Rabinowitz Floer homology is a locally linearly compact vector space in the sense of Lefschetz, or, equivalently, a Tate vector space in the sense of Beilinson-Feigin-Mazur. Poincar\'e duality and the graded Frobenius algebra structure on Rabinowitz Floer homology then hold in the topological sense. Along the way, we develop in a largely self-contained manner the theory of linearly topologized vector spaces, with special emphasis on duality and completed tensor products, complementing results of Beilinson-Drinfeld, Beilinson, Rojas, Positselski, and Esposito-Penkov.