duff In this paper we are concerned with the four functors Y., Y3 if, *, and *, which map from the category of topological spaces X to the category diff of Q-spectra. The functor 3*( ) (or ?if ( )) maps the space X to the Qspectrum of stable topological (or smooth) pseudoisotopies of X. (The spaces diff dif 3i(X), id9 (X), i > 0, are Hatcher's deloopings of 970(X), % Yd(X) (cf. [29])). The functor J*( ) maps a path-connected space X to the algebraic Ktheoretic S2-spectrum for the integral group ring Z7r1X. (The spaces X(X), i > 0, are the Gersten-Wagoner deloopings of Z%O(X); cf. [27, 52].) 5*(X) is also defined on a nonpath-connected space X to be () (X1), where {Xi: i e I} are the path components of X and iI (Xi) indicates the direct limit of all finite products of the {I (Xi): i E I}. If X is path connected then the functor *-' ( ) maps X to the L -surgery classifying spaces for oriented surgery problems with fundamental group r I1X identified by their fourfold periodicity k 90(X) = k+4j (X) (cf. [35, 39, 44, 47]). In addition, if X is not path connected then we set Y77(X) = tiEI5*fJ'(Xi), where {Xi: i E i} are the path components of X. Results obtained by the authors over the past five years (cf. [15-25] and, in particular, [19; 16, A.1, A.18]), together with many earlier results and conjectures which are reviewed in 1.6.1-1.6.6, suggest that, for any connected CWcomplex X and for 5* () denoting any of the above functors, the Q-spectrum *(X) should be computable in a simple way from the values taken by 9'*() on covering spaces of X, which have very simple fundamental groups. In more detail, we let 9'(X) denote the collection of all subgroups H c ir IX, which are either finite or virtually infinite cyclic. A group H is virtually infinite cyclic if there is a short exact sequence of groups 1 -+ Z -+ H -+ F -+ 1 with F equal to a finite group and Z equal to the infinite cyclic group. For each such group we let XH -+ X denote the covering space projection corresponding to H. Then the Q-spectrum 5? (X) should be computable in a simple way from the Q-spectra {5 (XH): H E 9 (X)}. The purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any X

/pdf/isomorphism-conjectures-in-algebraic-k-theory-49s1ahgc87.pdf

Isomorphism conjectures in algebraic $K$-theory

Three types of manifolds, spherical, fiat, and hyperbolic, are paradigms of geometric behavior. These are the Riemannian manifolds of constant positive, zero, and negative sectional curvatures, respectively. (The positive and negative sectional curvatures may be assumed, after scaling, to be + 1 and -1.) There is an equivalent synthetic geometric definition of them in terms of coordinate charts for a manifold M. Namely, a spherical, flat, or hyperbolic structure on M is an atlas of coordinate charts {(Un' .t;,)lo: E J} for M, where each .t;, ( Un) is an open subset of the (unit) sphere Sn , Euclidean space R n , or (real) hyperbolic space H n , respectively, such that

/pdf/a-topological-analogue-of-mostow-s-rigidity-theorem-1j5o83nh37.pdf

A topological analogue of Mostow’s rigidity theorem

On homotopy invariance of higher signatures

We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.

/pdf/algebraic-k-theory-over-the-infinite-dihedral-group-an-3km8bm45v8.pdf

Algebraic K-theory over the infinite dihedral group: an algebraic approach

Two types of Nil-groups arise in the codimension 1 splitting obstruction theory for homotopy equivalences of finite CW‐complexes: the Farrell‐Bass Nil-groups in the nonseparating case when the fundamental group is an HNN extension and the Waldhausen Nil-groups in the separating case when the fundamental group is an amalgamated free product. We obtain a general Nil-Nil theorem in algebraic K ‐theory relating the two types of Nil-groups. The infinite dihedral group is a free product and has an index 2 subgroup which is an HNN extension, so both cases arise if the fundamental group surjects onto the infinite dihedral group. The Nil-Nil theorem implies that the two types of the reduced f ‐groups arising from such a fundamental group are isomorphic. There is also a topological application: in the finite-index case of an amalgamated free product, a homotopy equivalence of finite CW‐complexes is semisplit along a separating subcomplex.

/pdf/algebraic-k-theory-over-the-infinite-dihedral-group-an-43zo3s46xs.pdf

Algebraic K -theory over the infinite dihedral group: an algebraic approach

Manifolds with π 1 = G x α T

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Manifolds with π 1 = G x α T