Algebraic Topology, Homotopy and Homology by R. Switzer

By R. Switzer

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Extra info for Algebraic Topology, Homotopy and Homology

Sample text

We obtain other very simple automor­ phi c functions if we con sider a mapping on to the hal fplan e ~ «() B contained in the disk Izi < 1, >0 of a domain where B is an arbitrary circular triangle all sides of which are orthogonal to the circle Izl = 1 and all interior angles of which are nonZero. If the interior angles of this triangle have values TTlm l , TTlm 2 , and "1m3, where m l , m2 , and m3 are positive integers, then, just as in the case of modular functions, the function (= ¢(z), which maps the domain B univalently onto the halfplane ~ «() > 0, can be extended to the entire disk Iz I < 1.

If an Tn (z) is obviously obtained from the equation I:=ll T; (a)1 Izl "" if a, {3, and y coincide, the triangles must be constricted to a point. p lie in K n . ' borhood of zo' If a family 'iJl "" l[Cz)\ is normal in a domain B, it is obviously normal at each point Zo in B. Let us prove the opposite, namely that if a family 'zl < 1 belongs to one of the 00. that there exists a point z I in the disk z II < 1 that does not belong to any tri­ angle of the grid. Let us draw a line segment from z I to some point of the domain center at B, We conclude that this segment contains points of an infinite set of triangles of the grid since it would otherwise be possible to get from B to zl by means of a only on the boundary of B.

Therefore, the function f(tk+ dC ) - f(tk) d· f' (ek) = This, itl~conjuction with what we have already proved, completes the Let zl and z2 denote any two points in E. In the domain B', we can find an = Z2 such that any two consecutive points Izl proof of the theorem. (11) for this domain, it suppose that E is a closed bounded domain. Let d denote the distance between ';n and z2' If we reverse the roles show that this ifl~quality also holds for arbitrary z 1 and z 2 in the domain must also hold (with the same M) for any set E contained in it.

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