## Elements of Combinatorial and Differential Topology |

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Elements of Combinatorial and Differential Topology Viktor Vasilʹevich Prasolov Limited preview - 2006 |

### Common terms and phrases

assume barycentric subdivision base point belongs Borsuk-Ulam theorem boundary cell choose circle Clearly coincides completely labeled connected components Consider the map constant map construct contained continuous map coordinates corresponding critical points CW-complex cycle defined deleted denote determined diffeomorphism dimension disjoint disk domains easy to verify edges element embedding exists fibration follows function f fundamental group graph G Hausdorff Hausdorff space hence homeomorphic homotopy equivalent hyperplane identity map intersection isomorphic Jacobian joined lemma loop manifold Mn map f map g map h matrix Mobius band Morse function neighborhood obtain one-to-one open cover open set orientation orthogonal pairwise path path-connected planar graph plane point xq polyhedron polynomial preimage Problem projection Proof prove pseudomanifold simplex simplicial complex singular points smooth map Sn~l Sperner's lemma sphere subgroup subspace Suppose tangent vector Theorem topological space triangulation two-dimensional surface vector field vertex vertices

### Popular passages

Page 8 - Kuratowski [5] proved the classic theorem that a graph is nonplanar if and only if it contains a subgraph homeomorphic to K$ or K*,i.

Page 13 - A given planar graph is outerplanar iff it can be embedded in the plane in such a way that all nodes lie on an external infinite face.

Page 1 - A, and the interior of A is the union of all open sets contained in A. The most important example of a topological space is the Euclidean space Rn. The open sets in R" are the balls D%£ = {xe R" : ||z - a|| < e} and all their unions.

Page 2 - If X is a topological space and Y is a subset of X, then Y can be endowed with the induced topology, which consists of the intersections of Y with open subsets of X. This turns the sphere 5" = {x € K"+1 : \\x\\ = 1} into a topological space.