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

TL;DR

This paper proves that three specific combinatorial 8-manifolds with quaternionic projective plane cohomology cannot be embedded in 12-dimensional space, despite having tight embeddings in 14-dimensional space, extending known nonembedding methods.

Contribution

It demonstrates nonembeddability of certain combinatorial 8-manifolds with quaternionic projective plane cohomology in 12-space, expanding embedding theory methods.

Findings

None of the three manifolds embed in Euclidean 12-space.

They have tight polyhedral embeddings in Euclidean 14-space.

The result extends methods used for real and complex projective planes.

Abstract

In this paper we prove, that none of the three combinatorial 8- manifolds on 15 vertices constructed by Brehm and Kuhnel, each of which is a cohomology quaternionic projective plane, can be combinatorially embedded in the Euclidean 12-space, though they have tight polyhedral embeddings in Euclidean 14-space. This extends a similar method already known for the nonembeddings of real and complex projective planes.