Twisted Dirac operators on certain nilmanifolds associated to even lattices

TL;DR

This paper constructs twisted Dirac operators on nilmanifolds derived from even lattices, computes their eta-invariants in the adiabatic limit, and applies these results to analyze elements in stable homotopy groups.

Contribution

It introduces a new geometric setup involving nilmanifolds from even lattices and computes eta-invariants to study stable homotopy classes.

Findings

Reduced eta-invariant computed in the adiabatic limit

Identification of elements in stable homotopy groups via eta-invariants

Application to the Adams-Novikov spectral sequence

Abstract

Starting from an even definite lattice, we construct a principal circle bundle covered by a certain three-step nilpotent Lie group G. On the base space, which is again a nilmanifold, we then study the Dirac operator twisted by the associated complex line bundles. Noting that the whole situation fibers over the circle, we are able to determine the reduced eta-invariant of these Dirac operators in the adiabatic limit. As an application, we consider the total space of the circle bundle, equipped with a parallelism induced by G, as an element in the stable homotopy groups of the sphere and use the eta-invariants to analyze its status in the Adams-Novikov spectral sequence.