Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

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We study the homeomorphism types of manifolds h‐cobordant to a fixed one. Our investigation is partly motivated by the notion of special manifolds introduced by Milnor in his study of lens spaces. In particular, we revisit and clarify some of the claims concerning h‐cobordisms of these manifolds.

Topological free involutions on S 1 × S n are classified up to conjugation. We prove that this is the same as classifying quotient manifolds up to homeomorphism. There are exactly four possible homotopy types of such quotients, and surgery theory is used to classify all manifolds within each homotopy type. Mathematics Subject Classification (2000)57S25 · 57R67 · 55P10

Recent computations of UN il-groups by Connolly, Ranicki and Davis are used to study splittability of homotopy equivalences between 4-dimensional manifolds with infinite dihedral fundamental groups.The problem of splitting a manifold into a connected sum is one of the most natural, yet one of the most difficult problems in manifold topology. It was extensively studied by S. Cappell [4][5][6][7][8][9], who also provided an elegant solution. To be more specific: let f : M n → X n = X n 1 #X n 2 , n 5, be a homotopy equivalence of closed topological manifolds, where X n is a connected sum of manifolds X n i , i = 1, 2. The homotopy equivalence f is splittable if it is homotopic to a map (necessarily a homotopy equivalence which we continue to call f ), transverse regular to the separating sphere S n −1 ⊂ X n , such that the restriction of f to Y = f −1 (S n −1 ), and the two components of f −1 (X n − S n −1 ) are homotopy equivalences.Cappell's solution to the splitting problem for f : M n → X n is in terms of certain exotic UN il groups introduced and studied by him. Despite the fact that these groups are very difficult to deal with, he was able to perform various computations which lead to many important topological results ([7, 8]). The simplest example of a group G for which UN il groups can be nontrivial is that of an infinite dihedral group D ∞ % Z/2 * Z/2. In this case, Cappell was able to show that UN il 0 (Z/2 * Z/2) = 0 and that UN il 2 (Z/2 * Z/2) contains an infinite dimensional vector space over Z/2 (cf. [11]. Here we use the notation UN il * (Z/2 * Z/2) = UN il * (Z; Z, Z)). The odd-dimensional UN il groups UN il 1 (Z/2 * Z/2) and UN il 3 (Z/2 * Z/2) resisted calculations for quite some time. (It was known, however ([13]), that in this case the UN il-groups are either trivial or infinitely generated.) Recently F. Connolly and A. Ranicki, using rather complicated algebraic machinery, have shown that UN il 1 (Z/2 * Z/2) = 0 and UN il 3 (Z/2 * Z/2) contains an infinite dimensional vector space over Z/2 [12], and † Supported in part by NSF grants INT-9505890 and INT-98115