By Massey

This booklet is meant to function a textbook for a direction in algebraic topology firstly graduate point. the most subject matters coated are the category of compact 2-manifolds, the basic staff, overlaying areas, singular homology thought, and singular cohomology concept. those issues are constructed systematically, fending off all unecessary definitions, terminology, and technical equipment. anyplace attainable, the geometric motivation in the back of a number of the innovations is emphasised. The textual content involves fabric from the 1st 5 chapters of the author's previous ebook, ALGEBRAIC TOPOLOGY: AN advent (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY thought (GTM 70). the cloth from the sooner books has been rigorously revised, corrected, and taken brand new.

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27rZ (2) k(s)u-Sds is given by the formulae o 1 Jl { k(u;x,y) = ylog t A u log x2 ° if if if if u> y2 xy < u < y2 x 2 < u < xy u < x 2. Chapter 2 Artin L- Functions 54 Now consider the integral 1. JK = -2 7rZ r (- ;k (s)) k(s; x, y)ds. "K On the one hand, it is equal to (logYlx)2 - Lk(p;x,y) p where p runs over all zeroes of (K(S). Write p = (3 + i'y. 2]) NK(r; so) « 1 + r(log IdKI + nK log(lsol + 2)). Since it follows that L k(p; x, y) «x- 2c5 ~~1-c5 1 00 1 r2dNK(r; 1) c5 «x- 2c5 (8- 2 + 8- 1 log IdKI).

2, we have 7fD(X) = II~: Lix + O(IDI~x~nF logM(K/F)x). Proof We have 7fD(X) IDI. ICI LIX . ) -1Gf LIX = '" ~ ( 7fc(x) -1Gf c where the sum is taken over all conjugacy classes C contained in D. 2. Remark. 1 as O(IClx~nF logM(K/F)x). Thus Artin's conjecture allows us to replace ICI with ICI ~. 1 even without assuming Artin's conjecture. We give two such results below. Chapter 2 Artin L- Functions 50 Let D be a union of conjugacy classes in C and let H be a subgroup of C satisfying (i) Artin's conjecture is true for the irreducible characters of H (ii) H meets every class in D.

This has the following immediate corollary. If KIF is a Galois extension of odd degree and (K (s) has a zero of order:::; 3 at a point So then all Arlin L-functions of KIF are analytic at so. 2 of Stark. Of course, Stark's result makes no assumption on the Galois group of KIF. We give a brief outline of the proof. Assume the theorem is false, and take G to be a minimal counterexample for which Artin's conjecture fails, at a point s = So where the order of (K(S) is small as explained in the statement.