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The Brown-Peterson cohomology for skeleta of the classifying space of the group Z 4 ¢ 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.doi:10.18910/12578 fatcat:niqkpec4svbsfdghykl6l7sray