# Geometry and Arithmetic

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#### pp: 23–56

#### DOI: 10.4171/119-1/2

^{[1]}and Fabrizio Catanese

^{[2]}(1) Universität Bayreuth, Germany

(2) Universität Bayreuth, Germany

We show that a family of minimal surfaces of general type with $p_g = 0, K^2=7$, constructed by Inoue in 1994, is indeed a connected component of the moduli space: indeed that any surface which is homotopically equivalent to an Inoue surface belongs to the Inoue family.

The ideas used in order to show this result motivate us to give a new definition of varieties, which we propose to call Inoue type manifolds: these are obtained as quotients $ \hat{X} / G$, where $ \hat{X} $ is an ample divisor in a $K(\Gamma, 1)$ projective manifold $Z$, and $G$ is a finite group acting freely on $ \hat{X} $. For these types of manifolds we prove a similar theorem to the above, even if weaker, that manifolds homotopically equivalent to Inoue type manifolds are again Inoue type manifolds.

*Keywords: *Moduli of surfaces, surfaces with *p _{g}* = 0, group actions, topological methods