BP-theoretic instabilities to the motion planning problem in 4-torsion lens spaces

Abstract

The Brown-Peterson cohomology for skeleta of the classifying space of the group $\mathbb{Z}_{4}\times\mathbb{Z}_{4}$ is analyzed in order to describe obstructions to the motion planning problem for a particle moving in a 4-torsion lens space. We discuss the relationship of this situation to the Euclidean immersion problem for $2^{e}$-torsion lens spaces, and the way this leads to an alternative approach to the classical immersion problem for real projective spaces.