PROOF. We first observe that the pair (M, dM) is (m — k — 1)connected. Indeed, since inclusion induces an isomorphism of fundamental groups, we can use the relative Hurewicz theorem to compute the first nonvanishing relative homotopy group: TTi(M, dM) =TTi(My dM) (where M denotes the universal cover), 7Ti(M, dM) Ç^Hi(M, dM)=HTM~\M), by duality, where c denotes compact cohomology, and H?~\M)^HTM...

WHEN the elliptic plane is derived from a sphere by identifying antipodal points, the five convex regular solids inscribed in the sphere yield five regular tessellations of the elliptic plane: arrangements of equal regular polygons fitting together to fill the plane without interstices. The actual cases are exhibited in Table I, where the last column indicates that, since the tetrahedron lacks ...

The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Grünbaum–Dress polyhedra (that is, the regular apeirohedra, or apeirotopes of rank 3) in ordinary space E3 in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to describe all the discrete regular apeirotopes of...

made a study of honeycombs whose cells are equal regular polytopes in spaces of positive, zero, and negative curvature. The spherical and Euclidean honeycombs had already been described by Schlaf li (1855), but the only earlier mention of the hyperbolic honeycombs was when Stringham (1880, pp. 7, 12, and errata) discarded them as "imaginary figures", or, for the two-dimensional case, when Klein...