Special Holonomy on Special Spaces

TL;DR

This paper classifies certain 7-dimensional manifolds with special holonomy, focusing on biquotients and rational ellipticity, and explores their geometric and topological properties.

Contribution

It characterizes biquotients that may admit G_2 holonomy metrics and classifies rationally elliptic homotopy types of such manifolds.

Findings

At most three rational homotopy types of G_2-manifolds among biquotients.

All identified rationally elliptic manifolds are formal.

Classification of 7-dimensional rationally elliptic biquotients.

Abstract

We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds---all of them being formal. In the course of this examination we classify rationally elliptic homotopy types and characterise 7-dimensional simply-connected biquotients from a rational point of view. Moreover, we also investigate further manifolds of special holonomy, like manifolds of holonomy Spin(7) or Sp(n)\Sp(1) in special situations provided by rational ellipticity or geometric formality.