By Joseph J. Rotman

**Publish 12 months note:** First released in 1988

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A transparent exposition, with routines, of the fundamental principles of algebraic topology. compatible for a two-semester path before everything graduate point, it assumes an information of element set topology and uncomplicated algebra. even though 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 ideas as they're brought.

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

**Example text**

Pm. Let S denote the set of all convex combinations. [Po, ... , Pm] C s: It suffices to show that S is a convex set containing {Po, ... , Pm}· First, if we set tj = 1 and the other t; = 0, then we see that Pj E S for every j. : 0 and La; = 1 = Lb;. We claim that ta + (1 - t)f3 E S for tEl. Now PROOF. ta + (1 m - t)f3 = L Eta; ;=0 + (1 - t)bJp;. This is a convex combination of Po, ... : 0 because each term is nonnegative. S C [Po, ... , Pm]: If X is any convex set containing {Po, ... : O. If m = 0, then S = {Po} and we are done.

29. For n ~ 1, show that S" is a deformation retract ofR"+1 - {O}. 30. For n ~ 1, show that S" is a deformation retract of the "punctured disk" D"+1 - {O}. 1. 31. Let a = (0, ... , 0, 1) and b = (0, ... , 0, -1) be the north and south poles, respectively, of S·. Show that the equator S·-l is a deformation retract of S' - {a, b}, hence S·-l and S' - {a, b} have the same homotopy type. 32. Assume that X, Y, and Z are spaces with X c Y. If X is a retract, then every continuous map f: X --+ Z can be extended to a continuous map l: y --+ Z, namely, = fr, where r: Y --+ X is a retraction.

O. If m = 0, then S = {Po} and we are done. Let m > O. : 0 and L t; = 1, is P = tiP; in X? We may assume that to #- 1 (otherwise P = Po E X); by induction, L q = C~ tJ Pl + ... + C~ tJ Pm E X (for this is a convex combination), and so P = toPo + (1 - to)q E X, because X is convex. 3. The affine set spanned by {Po, Pl' ... , Pm} eRn consists of all affine combinations of these points. PROOF. A minor variation of the proof just given. D Definition. An ordered set of points {Po, Pl' ... , Pm} C Rn is affine independent if {Pl - Po, P2 - Po, ...