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We consider locally standard 2-torus manifolds, which are a generalization of small covers of Davis and Januszkiewicz and study their equivariant classification. We formulate a necessary and sufficient condition for two locally standard 2-torus manifolds over the same orbit space to be equivariantly homeomorphic. This leads us to count the equivariant homeomorphism classes of locally standard 2-torus manifolds with the same orbit space.
Euler's polyhedron theorem states for a polyhedron p, thatV - E + F = 2,where V, E, and F are, respectively, the number of vertices, edges, and faces of p. The formula was first stated in print by Euler in 1758 [11]. The proof given here is based on Poincaré's linear algebraic proof, stated in [17] (with a corrected proof in [18]), as adapted by Imre Lakatos in the latter's Proofs and Refutations [15].As is well known, Euler's formula is not true for all polyhedra. The condition on polyhedra considered...
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