# Inoue type manifolds and Inoue surfaces: a connected component of the moduli space of surfaces with $K_{2}=7,p_{g}=0$

### Ingrid Bauer

Universität Bayreuth, Germany### Fabrizio Catanese

Universität Bayreuth, Germany

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## Abstract

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 $X^/G$, where $X^$ is an ample divisor in a $K(Γ,1)$ projective manifold $Z$, and $G$ is a finite group acting freely on $X^$ . 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.