Title: Compact Manifolds with Exceptional Holonomy

In the classification of Riemannian holonomy groups, the {\it exceptional holonomy groups} are $G_2$ in 7 dimensions, and $Spin(7)$ in 8 dimensions. We outline the construction of the first known examples of compact 7- and 8-manifolds with holonomy $G_2$ and~$Spin(7)$. In the case of $G_2$, we first choose a finite group $\Gamma$ of automorphisms of the torus $T^7$ and a flat $\Gamma$-invariant $G_2$-structure on $T^7$, so that $T^7/\Gamma$ is an {\it orbifold}. Then we resolve the singularities of $T^7/\Gamma$ to get a compact 7-manifold $M$. Finally we use analysis, and an understanding of Calabi-Yau metrics, to construct a family of metrics with holonomy $G_2$ on $M$, which converge to the singular metric on~$T^7/\Gamma$.

1991 Mathematics Subject Classification: 53C15, 53C25, 53C80, 58G30.

Keywords and Phrases: exceptional holonomy, $G_2$, $Spin(7)$, Ricci-flat.

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