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# New PDF release: Algebraic and geometric topology

By Symposium in Pure Mathematics Stanford University 1976, Visit Amazon's R. James Milgram Page, search results, Learn about Author Central, R. James Milgram, , American Mathematical Society

ISBN-10: 082181432X

ISBN-13: 9780821814321

ISBN-10: 0821814737

ISBN-13: 9780821814734

Includes sections on Algebraic \$K\$- and \$L\$-theory, surgical procedure and its purposes, team activities

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This quantity grew from a dialogue by way of the editors at the trouble of discovering solid thesis difficulties for graduate scholars in topology. even if at any given time we each one had our personal favourite difficulties, we said the necessity to provide scholars a much wider choice from which to decide on a subject matter odd to their pursuits.

This textbook in element set topology is geared toward an upper-undergraduate viewers. Its mild speed can be necessary to scholars who're nonetheless studying to write down proofs. must haves comprise calculus and a minimum of one semester of study, the place the coed has been appropriately uncovered to the guidelines of uncomplicated set conception akin to subsets, unions, intersections, and features, in addition to convergence and different topological notions within the actual line.

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Example text

Surgering the torus S ∪ A via D yields a sphere, which bounds a ball in M since M is irreducible. Hence S ∪ A bounds a solid torus and S is ∂ -parallel, being isotopic to A (rel ∂S ). 5. , transverse to all fibers. Proof: Let C1 , · · · , Cn (n ≥ 1) be fibers of the Seifert fibering which include all the multiple fibers, and let M0 be M with small fibered open tubular neighborhoods of all the Ci ’s deleted. M0 is then a circle bundle π : M0 → B0 over a compact surface B0 with ∂B0 = ∅ . Choose disjoint arcs in B0 whose union splits B0 into a disk (we may assume 32 Torus Decomposition Ch.

2) Eliminate arcs of S ∩ σ 2 with both endpoints on the same edge of σ 2 , by isotopy of S. After these simplifications, components of M || S meeting 2-simplices only in rectangles are I -bundles (disjoint from ∂M ), as before. Trivial I -bundles are ruled out by hypothesis (assuming n ≥ 2 , so that M is not a fiber bundle with fiber S = S1 ), while nontrivial I bundles are tubular neighborhoods of 1-sided (hence nonseparating) surfaces T1 , · · · , Tm , say. We may assume M is connected, and then reorder the components Si of S so that M | (S1 ∪ · · · ∪ Sk ) is connected and each of Sk+1 , · · · , Sn separates M | (S1 ∪ · · · ∪ Sk ) .

40 Seifert Manifolds Ch. 4 Chapter 4. Seifert Manifolds Seifert manifolds, introduced in the last chapter where they play a special role in the torus decomposition, are among the best-understood 3-manifolds. In this chapter our goal is the classification of orientable Seifert manifolds up to diffeomorphism. We begin with the classification up to fiber-preserving diffeomorphism, which is fairly straightforward. Then we show that in most cases the Seifert fibering is unique, so in these cases the diffeomorphism and fiber-preserving diffeomorphism classifications coincide.