By Torsten Wedhorn

This publication explains thoughts which are crucial in just about all branches of recent geometry similar to algebraic geometry, advanced geometry, or non-archimedian geometry. It makes use of the main obtainable case, actual and complicated manifolds, as a version. the writer specifically emphasizes the variation among neighborhood and international questions. Cohomology thought of sheaves is brought and its utilization is illustrated via many examples.

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Hint: Use that every homotopy of paths is uniformly continuous. 14. Let n 1 be an integer and let 0 ¤ v 2 Rn . Show that Rn n R 0 v is contractible. Deduce that S n 1 n fx0 g is contractible for any x0 2 S n 1 . 15. Let n 3 be an integer. Show that Rn n f0g and S n 1 are simply connected. 13), show that there exists 0 ¤ v 2 Rn such that R 0 v 2 Rn n f g. 14. 16. Show that a covering map of finite degree is proper. Show that a covering map of degree 1 is a homeomorphism. 17. Let n 1 be an integer.

Indeed, R is simply connected because it is convex. The map is a covering: For j D ` 1; 1 let Uj D S 1 nfj g. Then U 1 [U1 D S 1 . n; n C 1/ ! n 1 ; n C 12 /. 2 2. The function expW C ! C is a universal covering (use (1) and the polar decomposition of complex numbers). 3. For n 2, the map f W C ! C, z 7! f0g/ D 1 for all z0 2 U . Hence f is not a covering. 17). 30. The fibers of a covering map pW XQ ! 25). 27). Let pW XQ ! X and let f W Z ! X be a continuous map. A continuous map fQW Z ! XQ such that p ı fQ D f is called a lifting of f along p.

X; x/-action. We obtain a functor ˚x W (CovSp(X)) ! X; x/); p 7! 3) Here (Sets-G) denotes the category of sets together with a right action by the group G. x/ is transitive if and only if XQ is path connected. Indeed, the condition is clearly necessary. Conversely suppose that XQ is path connected. Q Then yQ D xQ Œp ı Q . x/ and choose a path Q in XQ from xQ to y. X; Q ! X; x/ consists of those homotopy classes of paths xQ of x Q x// Q whose lift to XQ with start point xQ has also end point x.