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On Some Problems From Combinatorial Topology
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145 b
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There is a close connection between Topology and Combinatorics stemming from the fact that most of the compact topological spaces can be build up with finite simplicial complexes. Though finding small triangulations of such spaces is a topic of active current interest, analogous questions about maps between them have, somewhat surprisingly, remained neglected. A smooth map of constant rank between two closed smooth manifolds is triangulable but it is usually a hard problem to determine the least number of vertices required to triangulate a manifold or a smooth map beyween manifolds. In this book we give minmals simplicial maps from 2-manifolds to the 2-sphere, simplicial maps from 3-sphere to the 2-sphere and simplicial branched coverings of the real projective plane. While constructing simplicial branched covering maps we could realize that certain topological branched coverings of the real projective plane do not exist and certain exists topologically but cannot be realized combinatorially. So simplicial constructions not only help us to understand the topological facts but also leads to new inventions.
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