math250-extprobs

math250-extprobs - Extra Problems for Math 250A Some of the...

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Unformatted text preview: Extra Problems for Math 250A Some of the (parts of) exercises below repeat or are special cases of results prove in class. If X is a Hausdor/ topological space then a covering space for X is a pair ( &;Y ) of Y a Hausdor/ topological space and & : Y ! X a surjective continuous map such that if p 2 X then there exists U an open neighborhood of p in X such that U is evenly covered by & . This means that & & 1 ( U ) = [ & 2 I U & with, U & open, U & \ U = ; if 6 = and & j U & is a homeomorphism of U & onto U . The cardinality of I is called the order of the covering if j I j = r then the covering is called an r-fold cover. 1. Prove that if M is a C 1 manifold and ( &;Y ) is a covering of M as a topological space then there exists an atlas f ( U ; & ) g 2 J for M such that each U is evenly covered by & . 2. Let M;&;Y be as in 1. and let f ( U ; & ) g 2 J be as in 1. Let & & 1 ( U ) = [ & 2 I U & be the disjoint union as in the de&nition of evenly covered. De&nebe the disjoint union as in the de&nition of evenly covered....
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math250-extprobs - Extra Problems for Math 250A Some of the...

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