By Carl B. Boyer, Uta C. Merzbach, Isaac Asimov

ISBN-10: 0471543977

ISBN-13: 9780471543978

Boyer and Merzbach distill hundreds of thousands of years of arithmetic into this interesting chronicle. From the Greeks to Godel, the math is marvelous; the solid of characters is extraordinary; the ebb and movement of rules is in every single place obtrusive. And, whereas tracing the advance of eu arithmetic, the authors don't forget the contributions of chinese language, Indian, and Arabic civilizations. absolutely, this is—and will lengthy remain—a vintage one-volume historical past of arithmetic and mathematicians who create it.

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Given an intersection graph / : T —* Y for C, a lifting map for / in X is a continuous map / ' : T -> X so that P(/'(T)) = /(T) for all 7 G I \ Note that given one lifting / ' there are others given by c o / ' where

Proof. We have a composition of coverings X - ^ X 'I F2, where p(x) = ~p(pj(x)) for all x £ X and lj is the Galois group for pj. Since ix,. is the inertia subgroup for Lj in the composition covering p, p is one to one over p~~1(Lj). Therefore, if V , ( C 1J C 2) is a n v element of G/Jj so that (1) ip(e\}e2)pj(f(ei)) and Pj(/'(e2)) lie one the same curve in p - 1 ( L j ) e (2) ^( i> ^2) is the image of ip(e\, e2) in G, then V , ( e i> e 2)/ / (ei) and /'(e2) lie on the same curve in p _ 1 (Lj).

9, there is a choice of lifting C" for each curve C in C in the covering p:X-+W2 so that (p,C)t->bipJc is lifting data for the C. ERIKO HIRONAKA 58 C. O r d e r i n g curves above C. To find the intersection matrix for the curves in p w l ( £ ) explicitly, we need to be able to order the curves in p~l(C) and find their intersection numbers. Recall that the curves in p~l(C) for any curve C in C are in one to one correspondence with cosets of the stabilizer subgroup He associated to C. Thus our goal now is to find the stabilizer subgroups explicitly.