Download e-book for iPad: A Primer on Riemann Surfaces by A. F. Beardon

By A. F. Beardon

ISBN-10: 0521271045

ISBN-13: 9780521271042

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Additional info for A Primer on Riemann Surfaces

Example text

In this way, every polyhedral surface is a Riemann surface. 31 2 1. Let discs and D^. holomorphic on each D be a disc with diameter Suppose that D . f : D -►

Show that [o] = Co} * C(O). Deduce that X is connected but not arcwise connected. 7 QUOTIENT SPACES Let If we give Y T be a topology on a topology, then a largest topology T^ on Y f X and let f : X Y be any function. may or may not be continuous. There is with respect to which, f is continuous: this is defined by B e Tf if and only if It is easy to check that T^ f 1 (B) e T. is a topology, that that this is the largest topology which makes the quotient topology induced by f f is continuous and continuous.

Let D = (t and let G be the group of transla­ tions z n- z + nX + my where X and y (m,n € Z) are linearly independent over 3R (that is, they lie on different lines through the origin). 1 is applicable so C/G is a Riemann surface. We shall show that topologically this is a torus. One approach is to construct the parallelogram with vertices 0,A,y and A+y and observe that corresponding points on opposite sides are in the same G-orbit. As the parallelogram contains at least one point from each orbit it seems likely that we can construct D/G by identifying the paired sides of the parallelogram.

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A Primer on Riemann Surfaces by A. F. Beardon

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