The Borsuk-Ulam theorem for closed 3-dimensional manifolds having Nil Geometry

Abstract

Let M be a closed, connected 3-manifold which admits Nil geometry, we determine all free involutions τ on M and the Borsuk-Ulam index of (M, τ ).

The Borsuk-Ulam theorem for closed $3$-manifolds having geometry $S^2\times \R$

“…(1) S The goal of the present work is to answer the problem above for the four Seifert manifolds of Remark 2, in the same spirit as we did for flat-and nil-geometries ([3], [4]). For most of the remaining Seifert manifolds 1 the same problem can be solved similarly, using [6] and [7].…”