Topology of Surfaces

1-cycle 2-complex 3-manifold abelian group Algebra antipodal points barycentric subdivision Calculus cell complex cellular function chain Chapter Ck(K Cl(A closed sets colors compact connected sum considered continuous function Corollary cube curve cylinder defined Definition deformation retract denote dimensions disc elements endpoints equivalence relation euler characteristic example Exercise f(vi faces finite number function f geometric glued graph H₁(K Hk(K Ho(K homeomorphic homology groups homotopy identified induces integer interior interval inverse k-cell k-chain Klein bottle Lemma limit point loop manifolds Mathematics Möbius band neighborhood Note number of edges number of vertices open sets orientable paths pictured in Figure planar diagram polygon projective plane Proof Prove Theorem rectangle regular complex Rn+1 sequence simplex simplicial approximation simplicial complex simplicial function subset T₁ topological property topological space topologically equivalent torus triangles vector field vertex νο ση υο