An Introduction to Algebraic Topology (Graduate Texts in by Joseph J. Rotman

By Joseph J. Rotman

Publish 12 months note: First released in 1988

A transparent exposition, with workouts, of the fundamental rules of algebraic topology. appropriate for a two-semester direction first and foremost graduate point, it assumes a data of element set topology and uncomplicated algebra. even supposing different types and functors are brought early within the textual content, over the top generality is refrained from, and the writer explains the geometric or analytic origins of summary innovations as they're brought.

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

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The affine maps on each tth interval give the formula for a function of two variables defined on the quadrilateral; this formula is used to show that this function is continuous. 44 3. The Fundamental Group Definition. Fix a point Xo E X and call it the basepoint. The fundamental group of X with basepoint Xo is 1t 1 (X, xo) = {[f]: [fJ is a path class in X with cx[fJ = Xo = w[fJ} with binary operation [fJ [g] = [f * g]. 3. 1t1 (X, xo) is a group for each Xo PROOF. E X. 2. D The Functor 11: 1 We have been led to the category Top*of pointed spaces and pointed maps that we introduced in Chapter O.

I) Define i: X --+ M f by i(x) = [x, 0] and j: Y --+ Mf by j(y) = [y]. Show that i and j are homeomorphisms to subspaces of M f. (ii) Define r: Mf --+ Y by rex, t] = f(x) for all (x, t) E X x I and r[y] = y. Prove that r is a retraction: rj = ly. (iii) Prove that Y is a deformation retract of M f . ) (iv) Show that every continuous map f: X --+ Y is homotopic to r 0 i, where i is an injection and r is a homotopy equivalence. 3 If A and B are topological spaces, then A II B denotes their disjoint union topologized so that both A and B are open sets.

Defmition. A subset X of R m is convex if, for each pair of points x, y E X, the line segment joining x and y is contained in X. In other words, if x, y E X, then tx + (1 - t)y E X for all tEl It is easy to give examples of convex sets; in particular, In, Rn, D n, and I1 n are convex. The sphere sn considered as a subset of R n+1 is not convex. Definition. A space X is contractible if Ix is nullhomotopic. 7. Every convex set X is contractible. 19 Convexity, Contractibility, and Cones PROOF. Choose Xo E X, and define c: X --+ X by c(x) = Xo for all x E X.

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