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Algebraic L-theory and Topological Manifolds (Cambridge Tracts in Mathematics): A. A. Ranicki

This book presents the definitive account of the applications of this algebra to the surgery classification of topological manifolds. The central result is the identification of a manifold structure in the homotopy type of a Poincaré duality space with a local quadratic structure in the chain homotopy type of the universal cover. The difference between the homotopy types of manifolds and Poincaré duality spaces is identified with the fibre of the algebraic L-theory assembly map, which passes from local to global quadratic duality structures on chain complexes. The algebraic L-theory assembly map is used to give a purely algebraic formulation of the Novikov conjectures on the homotopy invariance of the higher signatures; any other formulation necessarily factors through this one.

Summary: An excellent overview of generalized surgery theory

Rating: 5

For those readers who have completed advanced studies in algebraic topology and K-theory, and are ready to move on to a few of the even more esoteric topics in algebra and topology, this book would be an excellent choice. It is of course written for those who intend to specialize in the the areas included, but the book has value for those who are merely curious about the subject matter. The author introduces his book by describing it as a self-contained overview of the algebraic L-theory of quadratic forms in dimensions greater than or equal to 5. This theory was initiated over two decades ago by the mathematicians Browder, Novikov, Sullivan, and Wall for the case of compact, differentiable, and piecewise-linear manifolds, and was constructed to study the relationship between their topology and their homotopy type. This theory was then extended to the case of topological manifolds by the mathematicians Kirby and Siebenmann. The author does not assume the reader has any prior exposure to surgery theory. Readers though who have knowledge of the extreme difference in level of difficulty between doing topology in dimensions 3 and 4 versus in dimension 5 should find this book easier going, i.e. the "Whitney trick" holds in 5 dimensions (or above). The main core of the results in this book has its origins in the concept of Poincare duality, which is the equality between a spaces' n-dimensional homology and its (n-k)-cohomology. An n-dimensional Poincare space is a topological space where this equality holds for any coefficient group. Finite Poincare spaces, i.e. those which have the homotopy type of a finite CW complex, are not necessarily equivalent to a compact manifold. This motivates the "manifold structure existence" problem, which attempts to find out if a finite Poincare space (a space having the same homotopy type of a finite CW-complex) is homotopy equivalent to a compact manifold. But this homotopy equivalence may not be homotopic to a homeomorphism, and this fact motivates the "manifold structure uniqueness" problem, which is an attempt to find out if such a homotopy exists. A compact n-dimensional manifold is a finite n-dimensional Poincare space but a finite Poincare space is not necessarily homotopy equivalent to a compact manifold. The surgery theory of Browder, Novikov, Sullivan, Wall formulated obstructions for deciding the manifold structure existence and uniqueness theorems for dimensions greater than or equal to 5. These obstructions involve the topological K-theory of vector bundles, along with the algebraic L-theory of quadratic forms.In the book the author extends his earlier work on algebraic surgery and develops an intrinsic characterization of the manifold structures. Thus the structure groups are defined using algebraic Poincare complexes, these being obstructions to the existence and uniqueness problems. The total surgery obstruction is a homotopy invariant that vanishes if and only if the space is homotopy equivalent to a compact manifold (of dimension greater than or equal to 5). These kinds of ideas are then abstracted to yield the concept of an assembly map, which is a map that acts on a topological invariant and gives a homotopy invariant. The algebraic L-theory assembly map essentially measures how far a homotopy equivalence is from being homotopic to a homeomorphism, or, put another way, how far from being rigid a space is (a rigid manifold is one in which every homotopy equivalence is homotopic to a homeomorphism). The author discusses his notion of an assembly map as a generalization of other assembly maps in algebraic topology, and he gives examples: the Leray homology spectral sequence, and the Zeeman dihomology spectral sequences. The idea of assembly in these contexts is to piece together the homology of a space from its cohomology with coefficients in the local homology. For example, for the case of homology manifolds, where the local homology groups at a point are the local homology groups of Euclidean space, the local Poincare duality isomorphisms are "assembled" to the global duality isomorphisms. Another way of saying this is that the local homology at each point, which is a topologically invariant property is "assembled" into the homotopy invariant property of Poincare duality. The author calls the relationship between topological manifolds, Poincare spaces, local algebraic Poincare complexes, and global algebraic Poincare complexes a "fiber square" in the book. In analogy with hermitian K-theory, the quadratic L-groups were defined by the author in a previous work as cobordism of quadratic Poincare complexes over a ring with involution R. One then constructs an algebraic L-theory assembly map involving the generalized homology groups whose coefficients are taken from the quadratic L-theory spectrum of the integers. The structure groups are then taken to be the relative homotopy groups of the assembly map and they measure the extent to which the assembly maps fail to be an isomorphism. Thinking topologically, the computation of the structure set involves the use of the Spivak normal fibration. The structure set is nonempty only if there exists a stable vector bundle whose associated sphere bundle is isomorphic to the Spivak normal fibration. A finite Poincare space is homotopy equivalent to a compact iff the Spivak normal fibration admits a topological bundle reduction such that a normal map from a manifold to the Poincare space has zero surgery obstruction. Since the author's goal is both an algebraic and a topological setting, he defines an algebraic analog to the Spivak normal fibration. This analog involves first what the author calls an "algebraic bordism category", which is an additive category with chain duality A, and a pair of subcategories of the chain homotopy category of A. The author spends the first part of the book giving the details of how to construct the quadratic L-groups of this category, and then how to get the assembly functor. This algebraic theory is then related to the geometric/topological context in the second half of the book.

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