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In this paper we prove, that none of the three combinatorial 8- manifolds on15 vertices constructed by Brehm and Kuhnel, each of which is a cohomologyquaternionic projective plane, can be combinatorially embedded in the Euclidean12-space, though they have tight polyhedral embeddings in Euclidean 14-space.This extends a similar method already known for the nonembeddings of real andcomplex projective planes.

Combinatorial 8-Manifolds having Cohomology of the Quaternionic Projective Plane and Their Nonembeddings