Induction theorems for groups of homotopy manifold structures

Identifying assembly maps in K - and L -theory

“…One can now construct homotopy invariant functors from Spaces to Spectra by composing with the functor X → Rπ(X), where π(X) denotes the fundamental groupoid of a space equipped with the standard involution and R is a ring with unit. For L-theory we will see in § 10 that this functor agrees on homotopy groups in dimensions ≥ 5 with Wall's geometric definition [Wal70] of the surgery obstruction groups L n (Zπ(X)), and with Quinn's construction of the geometric surgery spectra L geom (X) (a full exposition of this construction has been given by Nicas [Nic82]), and also with the algebraic surgery spectra L alg (Zπ(X)) of Ranicki [Ran92a]. We will also see in § 4 that the Loday assembly map for K-theory can be recovered by this process.…”

“…for i ≥ 0 [Nic82]. In these dimensions both sides are almost 4-periodic, the difference being a factor of Z at i = 0 on the left-hand side.…”

A history and survey of the Novikov conjecture

“…Nicas [Ni1] used the abelian group structures to prove induction theorems for the structure set. If (M n , ∂M ) is an n-dimensional manifold with boundary and π 1 (M ) = Γ, then Siebenmann Periodicity [KirS] 15 shows that there is a monomorphism …”

Structure sets vanish for certain bundles over Seifert manifolds

“…Recall that the long exact sequence of topological surgery is an exact sequence of groups when [X, dX; G/TO?, *] is given the "characteristic variety" addition [17,20,21], and that the surgery invariant [Xn, dX; G/TOP, *] -+ Ln(irxX) is identified by Poincaré-Lefschetz duality with a natural transformation from the homology theory with coefficients in an ß-spectrum L0, whose 0th term is homotopy equivalent to G/TOP, to L-theory: o: h"(X)^ Ln(irxX). (This paper uses h"(X) for Hn(X;LQ).)…”

“…When K3 is sufficiently large this is Lemma 2.5. The remaining cases, in which the group irxK/j#(irxS1) is a hyperbolic triangle group, are dealt with by following Plotnick's induction scheme from §1 and applying part (3) of the following consequence of the induction theorems of Nicas [17] (see especially Proposition 6.2.9). Proof.…”