Buchstaber Invariants of Universal Complexes

Davis and Januszkiewicz introduced (real and complex) universalcomplexes to give an equivalent definition of characteristic maps ofsimple polytopes, which now can be seen as ``colorings''. The authorderives an equivalent definition of Buchstaber invariants of asimplicial complex $K$, then interprets the difference of the realand complex Buchstaber invariants of $K$ as the obstruction toliftings of nondegenerate simplicial maps from $K$ to the realuniversal complex or the complex universal complex. It was proved byAyzenberg that real universal complexes can not be nondegeneratelymapped into complex universal complexes when dimension is 3. Thispaper presents that there is a nondegenerate map from 3-dimensionalreal universal complex to 4-dimensional complex universal complex.