Floer homotopy type and eta invariants of Seifert $3$-manifolds fibering over $\mathbb{RP}^2$

TL;DR

This paper computes Floer homology, Floer homotopy types, and eta invariants for Seifert rational homology 3-spheres fibering over , revealing they are all L-spaces with simplified Floer homotopy types.

Contribution

It provides explicit calculations of Floer invariants and eta invariants for a class of Seifert 3-manifolds, extending understanding of their Floer homology and related invariants.

Findings

All studied manifolds are L-spaces.

Floer homotopy type is a suspension of S^0.

Explicit eta invariant formulas involving orbifold contributions.

Abstract

We compute the Floer homology and Seiberg-Witten Floer homotopy type of Seifert rational homology $3$-spheres which fiber over $\mathbb{RP}^2$. We show that they are all $L$-spaces and their Floer homotopy type is a suspension of $S^0$. Additionally, we compute the Ozsv\'ath-Szab\'o $d$-invariants, or equivalently the Seiberg-Witten $\delta$-invariants for such $3$-manifolds. This is done by computing the eta invariant of spin$^c$-Dirac operators associated to spin$^c$-connections covering the adiabatic connection, a certain metric connection distinct from the Levi-Civita connection. It turns out that this eta invariant involves a contribution given by the eta invariant of an orbifold pin$^c$-connection on the orbifold base of the Seifert fibration, which we also compute.