
Author(s)  Ranicki, A.A. 

Title  Algebraic LTheory and Topological manifolds 
Publisher  Cambridge University Press 
Year of publication  1992 
Reviewed by  VALERII V. TROFIMOV 
This textbook provides lucid coverage of the application of the abstract theory of quadratic forms to the surgery classification of the topological manifolds. In the mid 70s A.A. Ranicki completed the full formalization of the Hermitian algebraic Ktheory and gave the construction of the algebraic Ltheory. Thus, the readers get the book from the skillful master in this field.
A topological space X is called an ndimensional Poincare space if <formula>, where <formula> and <formula> are cohomology and homology groups respectivly with arbitrary coefficients. Two important problems in topology are considered in the book under review. First one is concerned the manifold structure existence problem, which is decided if a finite Poincare space is homotopy equivalent to a compact manifold. The second one is connected with the manifold structure uniqueness problem, which is determined if a homotopy equivalence of compact manifolds is homotopic to a homeomorphism, or at least hcobordant to one. Using BrowderNovikovSullivanWall surgery theory, the construction of the computable obstruction for deciding the manifold structure existence and uniqueness problems in dimension n, n >= 5, are given.
The book is devided into two parts. First part, Algebra (¤¤ 1  15 ), is elucidated the Ltheory of algebraic Poincare complexes in an additive category with chain duality, as well as algebraic bordism category. The second part, Topology ( ¤¤ 16  26), is devoted to the total surgery obstruction for solving two problems mentioned above.
This is unique book in this field and I can recommend one to postgraduate students in topology in the first place. Researchers in the number theory and algebra can find this book very valuable as well.
All in all, the scope of Algebraic LTheory and Topological Manifolds, ranges from postgraduated students to research level and its purchase is a good investment for readers with interest in abstract algebra and topology.