On the cohomology ring of flat manifolds with special structure

I.G.DOTTI AND R.J.MIATELLO

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

Unión Matemática Argentina

Lugar: Bahia Blanca; Año: 2006 vol. 46 p. 133 - 147

0041-6932

The purpose of the present paper is to study the real and complexcohomology ring of certain low dimensional hyperk"ahler compact flatmanifolds obtained by the "doubling" construction in [BDM]. We willdetermine the structure of this ring in the case of all $8$-dimensionalcompact flat manifolds which are obtained by doubling all$4$-dimensional K"ahler flat manifolds. Here weshall use the known classificationof space groups in dimension $4$ given in [BBNWZ]. As a second interesting class we shall study the manifolds obtained bydoubling (twice) a Hantzsche-Wendt type manifold.  This gives, for any$nge 3$, a $4n$-dimensional compact flat hyperk"ahler manifold withholonomy group ${f Z}_2^{n-1}$.  We shall show that this algebra isgenerated by the $G$-invariant forms of degree $2$ and $3$ and will findgenerators and relations for the cohomology ring.  In this case, this isnot a polynomial ring.