By Murnaghan F.D.

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The double of a dJak ill a sphere, and~ double of an annulus is a torus. §3. COVERING SURFACES The reader ia familiar with the crude notion of Riemann aurfaoes which in classical function theory. For the description of mob surfaces a pictorial language is used which in spite of ita di~c value does not belong in a mathematical treatment. The notion in question is purely topological, and in this section we are concerned with its a:Domatization under the name of covering surface. A covering surface is smooth if it has no branch points, and a smooth covering surface will be called regular if it lm!!

Ohaina into 0. (K). ). :)- I 21M where in general the right hand Bide needs to be :rewritten in ite simplest fonn. e. that identified chains have identified boundaries. For n;:a2 the verification is immediate, and these are the only climeosiona that ooour in this book. One finds by explicit computation that aas• .. 0 for every 'limplex. hole group Oa(K) into the zero element of Oa-a(K). JSD. A chain is called a cgcle if its boundary is zero. The n-climeDBional cycles form a subgroup Za(K} of Oa(K), and Za(K) is the kernel of the 60 I.

It ia often desirable to imbed a bordered surface in a surface. In this respect we prove: Theorem. Bvery borderetliiUrJaa 1' ccm be regularly imbetltkd in a IIUrJaa. If 1' u compad, it CXJn be regularly imbetltkd in a cloBed ~~Urjaa. We construct a standard imbedding which will find important use, especially in the theory of Riemann surfaces. , aml con11ider a. topological mapping fl' ofF onto Ft. In the sum F+F 1 we identify each p e B(F) with its image fl'(p), and topologize as in 2F. The resulting space ia denoted by 1'.

### The calculus of variations by Murnaghan F.D.

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