Joseph J. Rotman's An Introduction to Algebraic Topology (Graduate Texts in PDF

By Joseph J. Rotman

ISBN-10: 1461245761

ISBN-13: 9781461245766

Publish yr note: First released in 1988

A transparent exposition, with routines, of the fundamental principles of algebraic topology. compatible for a two-semester path before everything graduate point, it assumes a data of element set topology and simple algebra. even if different types and functors are brought early within the textual content, over the top generality is kept away from, and the writer explains the geometric or analytic origins of summary innovations as they're brought.

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Additional resources for An Introduction to Algebraic Topology (Graduate Texts in Mathematics, Volume 119)

Sample text

5. 1t o is not a very thrilling functor since its values lie in Sets, and the only thing one can do with a set is count it. Still, it is as useful as counting ordinary components (which is how one proves that S1 and I are not homeomorphic 28 1. Some Basic Topological Notions (after deleting a point)). ) of a sequence of functors. The next is 1t 1 , the fundamental group, which takes values in Groups; the others, 1t 2 , 1t 3 , ••• , are called (higher) homotopy groups and take values in Ab (we shall study these functors in Chapter 11).

8). (3) There is a category appropriate to the fundamental group functor 1t l' Define the pointed homotopy category, hTop*, as the quotient category arising The Functor 7tl 45 from the congruence of relative homotopy: if fo, fl: (X, x o) ~ (Y, Yo), then fo ~ fl rel{xo}· The objects ofhTop* are pointed spaces (X, x o), morphisms (X, x o) --+ (Y, Yo) are relative homotopy classes [f], where f: (X, x o) ~ (Y, Yo) is a pointed map, and composition is given by [h] [f] = [h 0 f] (when h, f can be composed in Top*).

If (X, xo) and (Y, Yo) are pointed spaces, then 1tl (X x Y, (xo, Yo)) ~ 1tl (X, xo) X 1tl (Y, Yo)· PROOF. Let p: (X x Y, (xo, Yo)) -+ (X, xo) and q: (X x Y, (xo, Yo)) -+ (Y, Yo) be the projections. Then (p*, q*): 1tl (X x Y, (xo, Yo)) -+ 1tl (X, xo) X 1tl (Y, Yo) is a homomorphism. In more detail, if f: I -+ X x Y is a closed path at (xo, Yo), then (p*, q*): [f] 1--+ (p* [f], q* [f]) = ([pf], [qf]). We show that (p*, q*) is an isomorphism by displaying its inverse. 5 shows that () is well defined.

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An Introduction to Algebraic Topology (Graduate Texts in Mathematics, Volume 119) by Joseph J. Rotman

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