By Roger Fenn

During this quantity, that's devoted to H. Seifert, are papers in line with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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Ce cours de topologie a été dispensé en licence à l'Université de Rennes 1 de 1999 à 2002. Toutes les buildings permettant de parler de limite et de continuité sont d'abord dégagées, puis l'utilité de l. a. compacité pour ramener des problèmes de complexité infinie à l'étude d'un nombre fini de cas est explicitée.

This publication is the 6th variation of the vintage areas of continuing Curvature, first released in 1967, with the former (fifth) variation released in 1984. It illustrates the excessive measure of interaction among crew idea and geometry. The reader will enjoy the very concise remedies of riemannian and pseudo-riemannian manifolds and their curvatures, of the illustration concept of finite teams, and of symptoms of contemporary development in discrete subgroups of Lie teams.

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Then P(u) is afundarnental domain of T, that is, no two distinct points of P(u) __ represent the same point of T, and the image of the closed parallelopiped P(u), under the covering, is all of T. Note that v01 T = V O P(u) ~ for every u E r. We simply refer to P(u) as a copy of T. Also note that if A: R" + R" is a linear transformation for which r = A(Z'), then r*= (A*)-'(E"),where A* is the adjoint of A. Therefore, if T* is the torus determined by r*then we have vol T = ldet A1 = (vol T*)-'; 3.

R, be pairwise disjoint normal domains in M , whose boundaries, when intersecting d M , do so transversally. Given an eigenvalue problem on M , consider, for each r = 1, . , m, the eigenvalue problem on R, obtained by requiring vanishing Dirichlet data on dR, n M and by leaving the 18 1. The Laplacian original data on aR, n aM unchanged. Arange all the eigenvalues of in an increasing sequence a,, . ,R, 0I v1 I I v2 * * a with each eigenvalue repeated according to its multiplicity, and let the eigenvalues of M be given as in (79).

From (44) we would conclude that t , E (0,n/2&). But 1(6) I A(n/2fi) = nK implies that { ( V c , - v's,)s"-'}(t) = {n(6) - n K } 1: Vs: is nonnegative on (0, 6). Therefore at t = t , we have V ( t , ) 2 0-a tradiction. Thus for K > 0, 6 > n/2<~, we have p(6) > 1(6). con- THEOREM4. If K = 0, that is, if M K= R", then there exist positive constants c D ,cNsuch that A(6) = Ci/62, p(6) = Ci/62 for all 6 > 0. PROOF:For K = 0 we have S,(t) = t, C,(t) = 1, so the dzerential equation under study is y" + ( n -t 1) ~ 1(1 + n - 2) y = 0.