By Richard Evan Schwartz

This ebook offers a couple of themes relating to surfaces, reminiscent of Euclidean, round and hyperbolic geometry, the elemental workforce, common protecting surfaces, Riemannian manifolds, the Gauss-Bonnet Theorem, and the Riemann mapping theorem. the most inspiration is to get to a few fascinating arithmetic with no an excessive amount of formality. The e-book additionally comprises a few fabric merely tangentially relating to surfaces, corresponding to the Cauchy tension Theorem, the Dehn Dissection Theorem, and the Banach-Tarski Theorem. The objective of the e-book is to give a tapestry of principles from a number of parts of arithmetic in a transparent and rigorous but casual and pleasant means. necessities comprise undergraduate classes in genuine research and in linear algebra, and a few wisdom of complicated research

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**Example text**

This 48 4. The Fundamental Group means that [e] plays the role of the identity element in π1 (Y, y0 ). Exercise 7. Let g be any loop. Deﬁne the loop g ∗ so that it satisﬁes the equation g ∗ (x) = g(1 − x). In other words, g ∗ traces out the same loop as g, but in the opposite direction. Prove the following result: If g1 and g2 are equivalent, then g1∗ and g2∗ are equivalent. Finally, prove that [g] ∗ [g ∗ ] = [e] and [g ∗ ] ∗ [g] = [e]. In other words, the inverse of [g] is given by [g ∗ ]. Combining Exercises 5, 6, and 7, we see that π1 (Y, y0 ) is a group.

7. 1. Let X be a surface. This means, ﬁrst of all, that X is a metric space. So, it makes sense to talk about open and closed sets on X and also continuous functions from X to other metric spaces. What makes X a surface is that each point x ∈ X has an open neighborhood U such that U is homeomorphic to R2 . You should picture U as a little open disk drawn around x. So X has the property that, around every point, it “looks like” the plane. 1. Exercise 9. The unit sphere S 2 in R3 is the set {(x, y, z)| x2 + y 2 + z 2 = 1}.

This 48 4. The Fundamental Group means that [e] plays the role of the identity element in π1 (Y, y0 ). Exercise 7. Let g be any loop. Deﬁne the loop g ∗ so that it satisﬁes the equation g ∗ (x) = g(1 − x). In other words, g ∗ traces out the same loop as g, but in the opposite direction. Prove the following result: If g1 and g2 are equivalent, then g1∗ and g2∗ are equivalent. Finally, prove that [g] ∗ [g ∗ ] = [e] and [g ∗ ] ∗ [g] = [e]. In other words, the inverse of [g] is given by [g ∗ ]. Combining Exercises 5, 6, and 7, we see that π1 (Y, y0 ) is a group.