By A. A. Ranicki

This publication provides the definitive account of the purposes of this algebra to the surgical procedure class of topological manifolds. The vital result's the id of a manifold constitution within the homotopy form of a Poincaré duality area with an area quadratic constitution within the chain homotopy form of the common conceal. the variation among the homotopy varieties of manifolds and Poincaré duality areas is pointed out with the fibre of the algebraic L-theory meeting map, which passes from neighborhood to international quadratic duality constructions on chain complexes. The algebraic L-theory meeting map is used to provide a merely algebraic formula of the Novikov conjectures at the homotopy invariance of the better signatures; the other formula unavoidably elements via this one.

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If in ex. (5,7) p* is an infrapolynomial, then p* is a factor of F* where N 2; co(z) = IT (z - z;), a=o z - Z,]' F*(z) = (,)(Z) CO(z) + azk+1I s=o + AN = 1, where a is a constant and O(z) where 2, > 0 for all j, 1o + 2 -{is a polynomial of degree less than n - N + 1 such that [co(z)q(z) + azm+1] is of degree not exceeding n [Shisha-Walsh 1]. Hint: Modify proof of Th. (5,2) by noting under Lem. (5,2) that c; = 0 if j = 1, 2, , k, m + 1, , n [Shisha 3]. 9. In ex. (5,7) the determination of an infrapolynomial p* is equivalent to that of finding the polynomial r" of class 5* 92 * = {ak+lZm-k-1 + ak+2Zm-k-2 + ...

This result provides further evidence of the invariant character of the polar derivative. EXERCISES. Prove the following. 1. If the zeros of a polynomial f (z) are symmetric in a line L, then between two successive zeros of f(z) on L lie an odd number of zeros of its derivative f(z) and any interval of L which contains all the zeros of f(z) lying on L also contains all the zeros of f'(z) lying on L. Hint: Apply (10,7) to Rolle's Theorem. 2. Let z = g(Z) be a rational function which has as its only poles those of , k.

Prove the following. 1. If the points a,k and b,2 lie in a convex region K, then in the region S(K, (vr - y)/(p + q)) lies at least one of the points z1 , z2 , rn zn which satisfy z; - a;1) z; - a,2) ... z; - a;,) = 0 (z, - b,1)(z; - b;2) ... ) Hint: Assume the contrary. 2. If all the points at which a given pth degree polynomial f (z) assumes n given values c1, c2 , , c,, are enclosed in a convex region K, and if the m; where the m, satisfy (8,2). are numbers satisfying (8,2), then all the points at whichf(z) assumes the average value c =I m2c;/im; =1 9=1 lie in the star-shaped region S(K, (ir - y)/p [Marden 7 and 8; for cases y = 0, Fekete 2 to 6 and Nagy 4].