Classical Topology and Combinatorial Group Theory by John Stillwell

By John Stillwell

In recent times, many scholars were brought to topology in highschool arithmetic. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formulation, and knots, the coed is ended in anticipate that those picturesque rules will come to complete flower in college topology classes. What a unhappiness "undergraduate topology" proves to be! In so much associations it's both a carrier path for analysts, on summary areas, in any other case an creation to homological algebra during which the single geometric job is the finishing touch of commutative diagrams. photographs are saved to a minimal, and on the finish the coed nonetheless does nr~ comprehend the best topological evidence, resembling the rcason why knots exist. for my part, a well-balanced advent to topology may still pressure its intuitive geometric point, whereas admitting the valid curiosity that analysts and algebraists have within the topic. At any fee, this is often the purpose of the current booklet. In help of this view, i've got the ancient improvement the place potential, because it sincerely exhibits the impact of geometric concept in any respect phases. this isn't to assert that topology got its major impetus from geometric recreations just like the seven bridges; particularly, it resulted from the l'isualization of difficulties from different components of mathematics-complex research (Riemann), mechanics (Poincare), and staff idea (Dehn). it really is those connec­ tions to different elements of arithmetic which make topology a tremendous in addition to a gorgeous topic.

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1. Show that the polygon 9 determined by a polygonal Jordan curve p may be triangulated, by first dividing it into convex polygons. 9. 2. Show that a polygonal arc does not separate R2. 3. Show that a semidisc (half 2-ball, cf. 7) may be separated by an arc. 3 0-graphs A figure °T consisting of a polygonal Jordan curve p and a simple polygonal arc P3 connecting points Q, S on p, and elsewhere lying in the interior of the polygon 9 determined by p, is called a 0-graph. If 9- is a 0-graph and pt, P2 denote the arcs into which p is divided by Q, S, then P3 separates an interior point Pt of ' pI from an interior point P2 Of P2 in 9.

Finally, one can give a combinatorial definition of homeomorphism using the notion of elementary subdivision. Two simplicial complexes are certainly homeomorphic if they possess isomorphic schemata (schemata which are identical up to renaming of vertices). More generally, they are homeomorphic if their schemata become isomorphic after finite sequences of elementary subdivisions, in other words, if they have a common simplicial refinement. We say that two complexes are combinatorially homeomorphic if this is the case.

Thus the problem of deciding when two presentations are the same, the isomorphism problem of Tietze 1908, is similar to the word problem-in both cases we can effectively enumerate the pairs of equal objects, and the difficulty is to find the pairs of unequal objects. It actually follows from basic results of recursive function theory (see Rogers 1967) that the two problems are of the same degree of unsolvability, that is, a solution of one would effectively yield a solution of the other. ) In individual cases, however, the isomorphism problem is usually harder to solve than the word problem.

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