By D.B. Fuchs, O.Ya. Viro, V.A. Rokhlin, S.P. Novikov, C. Shaddock

ISBN-10: 3642080847

ISBN-13: 9783642080845

ISBN-10: 3662105810

ISBN-13: 9783662105818

to Homotopy concept O. Ya. Viro, D. B. Fuchs Translated from the Russian by means of C. J. Shaddock Contents bankruptcy 1. uncomplicated options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four § 1. Terminology and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four 1. 1. Set conception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four 1. 2. Logical Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four 1. three. Topological areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five 1. four. Operations on Topological areas . . . . . . . . . . . . . . . . . . . . . . . . . five 1. five. Operations on Pointed areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight §2. Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. 1. Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. 2. Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. three. Homotopy as a course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eleven 2. four. Homotopy Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eleven 2. five. Retractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eleven 2. 6. Deformation Retractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2. 7. Relative Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thirteen 2. eight. k-connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thirteen 2. nine. Borsuk Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2. 10. CNRS areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2. eleven. Homotopy homes of Topological structures . . . . . . . . . . . 15 2. 12. ordinary crew buildings on units of Homotopy periods . . . . . . . . sixteen §3. Homotopy teams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 three. 1. Absolute Homotopy teams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 O. Ya. Viro, D. B. Fuchs three. 2. Digression: neighborhood structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 three. three. neighborhood platforms of Homotopy teams of a Topological house . . . . 23 three. four. Relative Homotopy teams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 three. five. The Homotopy series of a couple . . . . . . . . . . . . . . . . . . . . . . . . . 28 three. 6. Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 three. 7. The Homotopy series of a Triple . . . . . . . . . . . . . . . . . . . . . . . 32 bankruptcy 2. package suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 §4. Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 four. 1. normal Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 four. 2. in the neighborhood Trivial Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 four. three. Serre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 four. four. Bundles of areas of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 §5. Bundles and Homotopy teams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 five. 1. The neighborhood approach of Homotopy teams of the Fibres of a Serre package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Additional info for Topology II: Homotopy and Homology. Classical Manifolds

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Tr ), tr+1)[ep E Sphr(A, xon, where h is any xo-homotopy h : A x 1 ~ X from in : A ~ X to the constant map. Hence 1rr +1 (X, A, xo) ~ 1rr +1 (X, xo) x 1rr (A, xo). 7. The Homotopy Sequence of a Triple. Let (X, A, B) be a topological triple (that is, X is a topological space and BeA c X) with base point Xo E B. 4, the homotopy groups 1rr (X, A, xo), 1rr (X, B, xo), 1rr (A, B, xo) and homomorphisms in* : 1rr (A, B, xo) ~ 1rr (X, B, xo), rel* : 1rr (X, B, xo) ~ 1rr (X, A, xo) induced by the inclusions in : (A, B) ~ (X, B), reI: (X, B) ~ (X, A) are defined for r :::: 1.

A locally trivial bundle is a covering in the wide sense if its fibre is a discrete space. R, base Si and projection x t--+ e2:trix, and the bundle with total space and base Si and projection Z t--+ zm, where m is any non-zero integer. F. Since the projection of a product space onto a factor is an open map, the projection of a trivial bundle, and hence of a locally trivial bundle, is an open map. An obvious verification shows that the product of two trivial bundles is a trivial bundle, and the product of two locally trivial bundles is locally trivial.

1. Coverings. A. 2 that a covering in the wide sense is a 10calIy trivial bundle with discrete fibre. The total space of such a bundle is usually called a covering space. Clearly each point of a covering space has a neighbourhood that is mapped homeomorphically by the projection onto its image in the base. B. A covering in the wide sense is said to be a covering in the narrow sense or simplya covering ifthe covering space and the base are connected and non-empty. AU the fibres of a covering c1early have the same cardinality, caUed the number of sheets of the covering.

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Topology II: Homotopy and Homology. Classical Manifolds by D.B. Fuchs, O.Ya. Viro, V.A. Rokhlin, S.P. Novikov, C. Shaddock


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