We study CappelFs UNil group, UNÜ2 n (R\^\^-^ for any ring and pair of bimodules with Involution (R; ^I 9^_l ). We show that, in the geometrically significant cases, this group is isomorphic to the Wall-Ranicki L-group, £ e (A a [f]), for a certain additive polynomial extension category A a [f ]. We then introduce an Arf invariant for UNH 2n (R', R, R) when the involution is trivial. We use this to compute UNil^R', R, R) when R is a Dedekind domain in which 2 is prime. We also show that for a suitable choice of (A, a), the Nil group of (A, a) coincides with the Nil group of Bass-Farrell and with the Nil group of Waldhausen. 1991 Mathematics Subject Classification: 57N15, 57R67. § 1. Introduction 1.0 The primary goal of this paper is to analyze Cappell's group UNil* (R\ 3# l9 $ -±), n > 0 äs defined in [Cl], (See also [Rl] Chapter 7). In particular, we calculate UNÜ2" (R', R, R) for any semisimple ring, and any Dedekind domain in which 2 is prime. Our analysis inevitably also draws in the Nil groups defined by Waldhausen [Wl], [W2], [W3], (also see [C4] p. 125), by Farrell [Fl], and by Bass [B]. It leads to interesting new interpretations of these äs well. The main results are 2.11, 3.9,4.9, 6.1, and 6.2. We summarize our results in l .2 below. But first we want to explain why a better underStanding of UNil seems so important to us.1.1 Motivation. Suppose/: M -+ X is a homotopy equivalence of compact closed manifolds, and X is the union of two other manifolds, X=X i uX_ i9 with X 0 = dX t = dX_ x = n X_ 1 . Assume that X and X 0 are connected, and that the fundamental groups of X i9 X-l9 X, and X 0 are denoted G
We show that a tame ended stratified space X is the interior of a compact stratified space if and only if a K-theoretic obstruction γ * (X) vanishes. The obstruction γ * (X) is a localization of Quinn's mapping cylinder neighborhood obstruction. The main results are Theorem 1.6 and Theorem 1.7 below. In particular, this explains when a G-manifold is the interior of a compact G-manifold with boundary. Our methods include a new transversality theorem, Corollary 1.17.
Abstract. The problem of equivariant rigidity is the Γ-homeomorphism classification of Γ-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of Γ. In other words, this is the classification of cocompact E fin Γ-manifolds.We use surgery theory, algebraic K-theory, and the Farrell-Jones Conjecture to give this classification for a family of groups which satisfy the property that the normalizers of nontrivial finite subgroups are themselves finite. More generally, we study cocompact proper actions of these groups on contractible manifolds and prove that the E fin condition is always satisfied.
We show, for n ≡ 0, 1 (mod 4) or n = 2, 3, there is precisely one equivariant homeomorphism class of C 2 -manifolds (N n , C 2 ) for which N n is homotopy equivalent to the n-torus and C 2 = {1, σ} acts so that σ * (x) = −x for all x ∈ H 1 (N).If n ≡ 2, 3 (mod 4) and n > 3, we show there are infinitely many such C 2manifolds. Each is smoothable with exactly 2 n fixed points.The key technical point is that we compute, for all n ≥ 4, the equivariant structure set S TOP (R n , Γ n ) for the corresponding crystallographic group Γ n in terms of the Cappell UNil-groups arising from its infinite dihedral subgroups. 57S17; 57R67
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