By F. Thomas Farrell and L. Edwin Jones

Aspherical manifolds--those whose common covers are contractible--arise classically in lots of parts of arithmetic. They happen in Lie crew conception as definite double coset areas and in man made geometry because the house kinds holding the geometry. This quantity comprises lectures introduced by means of the 1st writer at an NSF-CBMS local convention on K-Theory and Dynamics, held in Gainesville, Florida in January, 1989. The lectures have been basically serious about the matter of topologically characterizing classical aspherical manifolds. This challenge has for the main half been solved, however the three- and four-dimensional instances stay crucial open questions; Poincare's conjecture is heavily with regards to the three-dimensional challenge. one of many major effects is closed aspherical manifold (of measurement $\neq$ three or four) is a hyperbolic area if and provided that its primary crew is isomorphic to a discrete, cocompact subgroup of the Lie workforce $O(n,1;{\mathbb R})$. one of many book's issues is how the dynamics of the geodesic circulate could be mixed with topological regulate conception to check correctly discontinuous crew activities on $R^n$. a few of the extra technical themes of the lectures were deleted, and a few extra effects acquired because the convention are mentioned in an epilogue. The ebook calls for a few familiarity with the cloth contained in a simple, graduate-level direction in algebraic and differential topology, in addition to a few common differential geometry.

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12. Cauchy's integral fonnula 55 This theorem remains true if from the interior L of C we omit one point z (or a finite number of such points), provided that lim (t - z) j(t) = (2) 0, t->-z when t tends to z by any path lying entirely inside C. For given a positive number s we can choose r = It - zl so small that Ij(t) I < sir, and so the modulus of the integral ofj(t) taken round the circle It-zl = r is less than 2ns. 12 C. The function j(t) satisfies (2) and is holomorphic everywhere within C except at z, where it is undefined.

Cauchy's integral fonnula 55 This theorem remains true if from the interior L of C we omit one point z (or a finite number of such points), provided that lim (t - z) j(t) = (2) 0, t->-z when t tends to z by any path lying entirely inside C. For given a positive number s we can choose r = It - zl so small that Ij(t) I < sir, and so the modulus of the integral ofj(t) taken round the circle It-zl = r is less than 2ns. 12 C. The function j(t) satisfies (2) and is holomorphic everywhere within C except at z, where it is undefined.

More generally we shall designate by L the region on the left of an observer who describes C in a prescribed sense. 11. The complex Stokes's theorem If the closed contour C is replaced by an open arc B D described in the sense B to D, the region L can often be conveniently defined by considering the arc as prolonged indefinitely by drawing tangents to it at Band D fig. 10 (ii), or by joining B to D by an arc to form a closed curve figs. 10 (iii), (iv). / 'D / / . _- ----~ ß Fig. 10 (ii) Fig. 10 (iii) Fig.