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Since its introduction by Friedhelm Waldhausen 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 Waldhausen'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.

An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S + T of two (L)-semigroups S and T having boundary B is the quotient space obtained from the union of S × { 0 } and T × { 1 } by identifying the point ( x , 0 ) in S × { 0 } with ( x , 1 ) in T × { 1 } for each x in B. It is shown that no (L)-semigroup sum of dimension less than or equal to five admits an H-space structure, nor does any (L)-semigroup sum obtained from (L)-semigroups having an Abelian boundary. In particular, such sums cannot be a retract of a topological group.

In this paper we prove a realizability theorem for Quinn’s mapping cylinder obstructions for stratified spaces. We prove a continuously controlled version of the s-cobordism theorem which we further use to prove the relation between the torsion of an h-cobordism and the mapping cylinder obstructions. This states that the image of the torsion of an h-cobordism is the mapping cylinder obstruction of the lower stratum of one end of the h-cobordism in the top filtration. These results are further used to prove a theorem about the realizability of end obstructions.

We apply concepts and tools from abstract homotopy theory to complex analysis and geometry, continuing our development of the idea that the Oka Principle is about fibrancy in suitable model structures. We explicitly factor a holomorphic map between Stein manifolds through mapping cylinders in three different model structures and use these factorizations to prove implications between ostensibly different Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. Going beyond the Stein case to a study of cofibrant models of arbitrary complex manifolds, using the Jouanolou Trick, we obtain a geometric characterization of an Oka property for a large class of manifolds, extending our result for Stein manifolds. Finally, we prove a converse Oka Principle saying that certain notions of cofibrancy for manifolds are equivalent to being Stein.

Using a group-theoretic construction due to Bestvina and Brady, we build ( n + 1 ) (n+1) -manifolds W W which admit partitions into closed, connected n n -manifolds but which do not have finite homotopy type.

Let p: E → B be a manifold approximate fibration between closed manifolds, where dim(E) ≥ 4, and let M(p) be the mapping cylinder of p. In this paper it is shown that if g: B × I → B × I is any concordance on B, then there exists a concordance G: M(p) × I → M(p) × I such that G∣ B × I = g and$G\\mid E \\times \\{0\\} \\times I = \\operatorname{id}_{E \\times I}$. As an application, if Nnand Mn + jare closed manifolds where N is a locally flat submanifold of M and n ≥ 5 and j ≥ 1, then a concordance g: N × I → N × I extends to a concordance G: M × I → M × I on M such that G∣ N × I = g. This uses the fact that under these hypotheses there exists a manifold approximate fibration p: E → N, where E is a closed (n + j - 1)-manifold, such that the mapping cylinder M(p) is homeomorphic to a closed neighborhood of N in M by a homeomorphism which is the identity on N.

A theorem is proved which generalizes both the Vietoris-Begle theorem and the cell-like theorem for spaces of finite deformation dimension. The proof is geometric and uses a double mapping cylinder trick.

A pl map is an approximate fibration if and only if all the maps of any iterated mapping cylinder decomposition are homotopy equivalences. This leads to a classifying space.

We characterize those compact subsets of the plane which have mapping cylinder neighborhoods, describe the neighborhood closures, and show that such neighborhood closures are topologically unique. The proofs employ the notion of prime ends. We also show that if UU is a mapping cylinder neighborhood of a pointlike continuum in S3{S^3}, then U¯\\overline U is a 33-cell.

We generalize Marshall Cohen’s notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let f:K→Lf:K \\to L be a proper simplicial map of locally finite simplicial complexes. The following are equivalent: (1) The dual cells of the map are all cones. (2) The dual cells of the map are homogeneously collapsible in KK. (3) The inclusion of LL into the mapping cylinder of ff is collared. (4) The mapping cylinder triad (Cf,K,L)({C_f},K,L) is homeomorphic to the product triad (K×I;K×1,K×0)(K \\times I;K \\times 1,K \\times 0) rel K=K×1K = K \\times 1. Condition (2) is slightly weaker than f−1{f^{ - 1}}(point) is homogeneously collapsible in KK. Condition (4) when stated more precisely implies ff is homotopic to a homeomorphism. Furthermore, the homeomorphism so defined is unique up to concordance. The two major applications are first, to develop the proper theory of “attaching one polyhedron to another by a map of a subpolyhedron of the former into the latter\". Second, we classify when two maps from XX to YY have homeomorphic mapping cylinder triads. This property turns out to be equivalent to the equivalence relation generated by the relation f∼gf \\sim g, where f,g:X→Yf,g:X \\to Y means f=grf = gr for r:X→Xr:X \\to X some transverse cellular map.