New PDF release: Geometric Topology in Dimensions 2 and 3

By Edwin E. Moise (auth.)

ISBN-10: 1461299063

ISBN-13: 9781461299066

ISBN-10: 146129908X

ISBN-13: 9781461299080

Geometric topology could approximately be defined because the department of the topology of manifolds which offers with questions of the life of homeomorphisms. in basic terms in really fresh years has this type of topology completed a sufficiently excessive improvement to receive a reputation, yet its beginnings are effortless to spot. the 1st vintage outcome was once the SchOnflies theorem (1910), which asserts that each 1-sphere within the aircraft is the boundary of a 2-cell. within the following few a long time, the main outstanding affirmative effects have been the "Schonflies theorem" for polyhedral 2-spheres in area, proved via J. W. Alexander [Ad, and the triangulation theorem for 2-manifolds, proved by way of T. Rad6 [Rd. however the so much notable result of the Nineteen Twenties have been adverse. In 1921 Louis Antoine [A ] released a unprecedented paper within which he four confirmed number of believable conjectures within the topology of 3-space have been fake. hence, a (topological) Cantor set in 3-space don't need to have a easily hooked up supplement; hence a Cantor set could be imbedded in 3-space in not less than primarily other ways; a topological 2-sphere in 3-space needn't be the boundary of a 3-cell; given disjoint 2-spheres in 3-space, there isn't unavoidably any 3rd 2-sphere which separates them from each other in 3-space; etc and on. the well known "horned sphere" of Alexander [A ] seemed quickly thereafter.

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Example text

It follows that {v 1, v 2, ••• , vn} and {w 1, w2 , ••• , wn} are linearly independent. Now for every v= n L i= I 26 aiviE Rn, 3 The Schonflies theorem for polygons in R2 we define n g(v)= ~ Now glan is the f of Theorem I. IX;W;. i= I D Let I be the interior of the polygon J in R2 . 2, i is a finite polyhedron IKI. If a 2 E K, and a 2 n J consists of one or two edges of a 2, then a 2 is free (in K). 1, I, 3, 4, and 7 are free, but 2, 5, and 6 are not. 1 Theorem 3. Let J be a polygon in R2 , let I be the interior of J, and let K be a triangulation of f.

Let J be a polygon in R2 . Then every homeomorphism f: J -1 can be extended to give a homeomorphism f': R2 - R2 . 6. -sphere (not necessarily a polygon) in R2, let U be a component of R2 - J, and let F = Fr U. ) Let v E F. -simplex vw such that vw - { v} c U. Then we say that v is linearly accessible from U. Some point of F is linearly accessible from U. 7. Let U and F be as in Problem 6. Then the set of all points of F that are linearly accessible from U is dense in F. ) 8. Let U and F be as in Problem 6.

The following definitions are now valid, because they are stated in such a way that they depend only on the equivalence classes [h], [g], and are independent of the choice of h and g in [h] and [ g]. (1) The support l[hJI of [h] is lhl. (2) If Tis a face of ah, then [hl-rJ is a face of [h], and h(-r)forms a face of [h]. (3) The dimension dim [h] of [h] is dim ah. (4) In each set l[hJI, the barycentric coordinates of a point v are those induced by h. (5) Given a Euclidean simplex ak, a PL simplex [h], and a mapping j: a k ~I[ hJl, if the barycentric coordinates of j( v) are linear functions of those of v, then!

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