By Hajime Sato

The only so much tough factor one faces while one starts to benefit a brand new department of arithmetic is to get a believe for the mathematical feel of the topic. the aim of this ebook is to assist the aspiring reader gather this crucial logic approximately algebraic topology in a quick time period. To this finish, Sato leads the reader via uncomplicated yet significant examples in concrete phrases. additionally, effects aren't mentioned of their maximum attainable generality, yet when it comes to the easiest and such a lot crucial circumstances. based on feedback from readers of the unique version of this publication, Sato has extra an appendix of invaluable definitions and effects on units, common topology, teams and such. He has additionally supplied references.Topics coated comprise primary notions resembling homeomorphisms, homotopy equivalence, primary teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute sessions. items and examples thought of within the textual content comprise the torus, the Mobius strip, the Klein bottle, closed surfaces, mobile complexes and vector bundles.

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**Example text**

Edges; e free vertex of e 3 • v e G is diagram 2 2 are inner v is a A graph with at least one edge, all of whose edges are inner, is bound to have a loop (since the number of edges is finite); it follows that a tree with at least one edge always possesses a non-inner edge. Removing a non-inner edge together with a free vertex of e, e of a graph, is called an elementary collapse, and a sequence of these is called a collapse. ) (We Since a collapse takes a tree to a smaller tree it is easy to prove that any tree collapses to a single vertex.

20 will be of considerable significance later when we come to calculate homology groups. 16. ~ It follows that all the Kirchhoff equations for loops are consequences of the equations for these basic loops. (2X) A basis for the i-cycles on a connected graph is a collection of 1-cycles of 1-cycle on G in terms of which every G can be expressed uniquely as a linear combina- tion with integer coefficients. is a basis. 20 But not every basis arises in this way from a maximal tree. For instance, it is impossible to find edges 1 2 3 e, e , e ,e 4 of the graph shown, such that the resulting cycles 1 2 3 4 z , z , z ,Z are the basis given by the boundaries of the four small equilateral triangles.

Being m G is connected there is a path from v ~ an integer. D. • PLANAR GRAPHS A planar graph is a graph which is isomorphic to a graph in 2 the plane JR. In this section we shall investigate some properties of planar graphs, mostly connected with the idea of cyclomatic number. At the end of the section we shall take up the question of whether our insistence that graphs have straight edges places any essential restriction on the class of planar graphs. Some proofs will be omitted from this section and others will not be given in full detail; this is in order to avoid devoting undue space to what is, in fact, a subject of periBesides, pheral interest to the study of homology theo~.