By Joseph J. Rotman

ISBN-10: 1461245761

ISBN-13: 9781461245766

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

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A transparent exposition, with routines, of the fundamental rules of algebraic topology. compatible for a two-semester path at the start graduate point, it assumes a data of aspect set topology and uncomplicated algebra. even though 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 techniques as they're brought.

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

**Example text**

Evaluate W(f) by rewriting f(t) = exp ](t) for some real-valued function] [exp s denotes e 2ltiS ]. With this rewriting, one can convert the line integral into an ordinary integral via the substitution z = f(t) = exp](t). Thus dz = z271:il'(t) dt and W(f) = -1. 2m f J -dz = z fl f'(t) - dt = f(l) - - f(O). 0 For example, let f(t) = e2ltimt be the function wrapping I around SI Iml times (counterclockwise if m ~ 0 and clockwise if m < 0). Here we may let ](t) = mt, and so W(f) = ](1) - ](0) = m. (Note that there are other possible choices for j, namely, ](t) = mt + k for any fixed integer k.

If X is a subset of R", then the affine set spanned by X in R" is the same as the affine set spanned by X in Rk. 3. ) + 2 points. (Hint: Definition. Let {Po, Pl' ... , Pm} be an affine independent subset of Rn, and let A be the affine set spanned by this subset. 4 gives a unique (m + I)-tuple (to, t l , ... , t m) with L ti = 1 and x = L:"=o tiPi. The entries of this (m + I)-tuple are called the barycentric coordinates of x (relative to the ordered set {Po, Pl' ... 2, the barycentric coordinates of a point relative to {Po, Pl' ...

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.

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

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