By Allan J. Sieradski

The therapy of the topic of this article isn't really encyclopedic, nor was once it designed to be appropriate as a reference handbook for specialists. fairly, it introduces the subjects slowly of their old demeanour, in order that scholars usually are not beaten by way of the last word achievements of a number of generations of mathematicians. cautious readers will see how topologists have progressively sophisticated and prolonged the paintings in their predecessors and the way such a lot stable principles achieve past what their originators expected. To motivate the improvement of topological instinct, the textual content is abundantly illustrated. Examples, too a number of to be thoroughly lined in semesters of lectures, make this article compatible for self sufficient examine and make allowance teachers the liberty to choose what they're going to emphasize. the 1st 8 chapters are appropriate for a one-semester direction ordinarily topology. the complete textual content is acceptable for a year-long undergraduate or graduate point curse, and offers a robust origin for a next algebraic topology path dedicated to the better homotopy teams, homology, and cohomology.

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**Example text**

Xn ) ∈ S n−1 : xk ̸= 0} and we define Ψk : P −1 (Vk ) −→ Vk × Z2 by Ψk (p) = Ψk (x1 , . . , xn ) = ([p], xk /|xk |). Then Ψk is a diffeomorphism of the form Ψk (p) = (P(p), ψk (p)), where ψk (p) = ψk (x1 , . . , xn ) = xk /|xk | and ψk (p · g) = (xk g)/|xk g| = (xk g)/|xk | = ψk (p)g. ,n is a trivializing cover of R Pn−1 and P Z2 → S n−1 −→ R Pn−1 is a principal Z2 -bundle over R Pn−1 . Remark: Since P −1 (Vk ) is a disjoint union of two open hemispheres on S n−1 (xk > 0 and xk < 0) each of which is mapped homeomorphically onto Vk by P, P : S n−1 −→ R Pn−1 is actually a covering space (page 81, [N4]).

Flat connections on trivial bundles) We let G → X × G −→ X be a trivial bundle, Θ the Cartan 1-form on G and π : X ×G −→ G the projection. As in Example #1, page 38, ω = π ∗ Θ is a connection form on X × G. The Maurer-Cartan equations (page 19) yield (page 329, [N4]) the equation of 36 1. , and from this it follows that 1 dΘ + [Θ, Θ] = 0, 2 1 dω + [ω, ω] = 0 2 (page 353, [N4]). Thus, flat connections ω are “flat” because their curvature forms Ω are identically zero. 2. (Natural connection on the complex Hopf bundle) Connections ω on P U (l)-bundles U (1) → P −→ X have a number of very special properties and we begin with a general discussion of a few of these.

Z n ) · g = (z 1 g, . . , z n g) described in Example #6, page 28. The orbit space is X = C Pn−1 and we let P : S 2n−1 −→ C Pn−1 be the quotient map. Then P(p · g) = P(p) for all p ∈ S 2n−1 and g ∈ U (1). For each k = 1, . . , n, let Vk = {[p] = [z 1 , . . , z n ] ∈ C Pn−1 : z k ̸= 0} and define Ψk : P −1 (Vk ) −→ Vk ×U (1) by Ψk (p) = Ψk (z 1 , . . , z n ) = ([p], z k /|z k |). ,n is a trivializing cover of C Pn−1 so P U (1) → S 2n−1 −→ C Pn−1 is a principal U (l)-bundle over C Pn−1 .