Andrew Ranicki

Novikov Conjectures, Index Theorems and Rigidity

This is an expository paper explaining coarse analogues of the Novikov Conjecture and describing how information on the original Novikov Conjecture can be derived from these. For instance, we will explain how Novikov’s theorem on the topological invariance of rational Pontrjagin classes is a consequence of a coarse theorem (whose proof we sketch in an appendix) that in turn also implies the Novikov Conjecture for nonpositively curved manifolds. We also formalize the technique so that it can be applied in a wide variety of other contexts. Thus, besides a few purely geometric results, we also discuss equivariant, A-theoretic, stratified, and foliated versions of the higher signature problem. Closely related papers are [GL], [CGM], [CP], [KaS], [HR], [Hu]. See also the surveys [We1], [FRW], for wider perspectives.

Novikov Conjectures, Index Theorems and Rigidity: A history and survey of the Novikov conjecture

1. Precursors of the Novikov Conjecture 8 Characteristic classes 8 Geometric rigidity 9 The Hirzebruch signature theorem 9 The converse of the signature theorem (Browder, Novikov) 10 Topological invariance of the rational Pontrjagin classes (Novikov) 11 Non-simply-connected surgery theory (Novikov, Wall) 11 Higher signatures 12 Discovery of special cases of the Novikov Conjecture (Rokhlin, Novikov) 13 2. The Original Statement of the Novikov Conjecture 13 O nerexennyh zadaqah 13 [An English Version:] Unsolved Problems 15 3. Work related to the Novikov Conjecture: The First 12 Years or So 17 Statements of the Novikov and Borel Conjectures 17 Mishchenko and the symmetric signature 18 Lusztig and the analytic approach 20 Splitting theorems for polynomial extensions 21 Cappell and codimension 1 splitting theorems 22 Mishchenko and Fredholm representations 23 Farrell-Hsiang and the geometric topology approach 24 Kasparov and operator-theoretic K-homology 25 Surgery spectra and assembly (Quinn) 25 4. Work related to the Novikov Conjecture: The Last 12 Years or So, I:

Projective surgery theory

A simple (resp. finite) n-dimensional Poincaré complex X (n ≥ 5) is simple homotopy (resp. homotopy) equivalent to a compact n-dimensional CAT (= DIFF,PL or TOP) manifold if and only if the Spivak normal fibration νX admits a CAT reduction for which the corresponding normal map (f, b) : M → X from a compact CAT manifold M has Wall surgery obstruction σ ∗(f, b) = 0 ∈ Ln(π1(X)) (resp. σ ∗ (f, b) = 0 ∈ Ln(π1(X))). The surgery obstruction groups L∗(π) (resp. L h ∗(π)) of a group π are defined algebraically as Witt groups of quadratic structures on finitely based (resp. f. g. free) Z[π]-modules, and geometrically as bordism groups of normal maps to simple (resp. finite) Poincaré complexes X with fundamental group π1(X) = π. The object of this paper is to extend the above theory to finitely dominated Poincaré complexes, that is Poincaré complexes in the sense of Wall [18], and to the Witt group L∗(π) of quadratic structures on f. g. projective Z[π]-modules introduced by Novikov [8], the groups denoted by U∗(Z[π]) in Ranicki [12]. A normal map (f, b) : M → X from a compact n-dimensional manifold M to a finitely dominated Poincaré complex X has a normal bordism invariant, the “projective surgery obstruction”

Noncommutative localisation in algebraic K-theory I

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic K-theory. The main result goes as follows. Let A be an associative ring and let A → B be the localisation with respect to a set σ of maps between finitely generated projective A-modules. Suppose that Tor A n (B,B) vanishes for all n > 0. View each map in a as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes D p e r f (A). Denote by the thick subcategory generated by these complexes. Then the canonical functor D p e r f (A) → D p e r f (B) induces (up to direct factors) an equivalence D p e r f (A)/ → D p e r f (B). As a consequence, one obtains a homotopy fibre sequence K(A,σ) → K(A) → K(B) (up to surjectivity of K 0 (A) → K 0 (B)) of Waldhausen K-theory spectra. In subsequent articles [26, 27] we will present the K- and L-theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of Tor A n (B,B), we also assume that every map in σ is a monomorphism, then there is a description of the homotopy fiber of the map K(A) → K(B) as the Quillen K-theory of a suitable exact category of torsion modules.

Controlled K-Theory

Abstract The controlled finiteness obstruction and torsion are defined using controlled algebra, giving a more algebraic proof of the topological invariance of torsion and the homotopy finiteness of compact ANRs.

Finite domination and Novikov rings

IN order to distinguish between the combinatorial properties of finite simplicial complexes and the topology of compact polyhedra and compact manifolds it is necessary to consider infinite simplicial complexes, non-compact polyhedra, open manifolds, and algebraic Kand L-theory. The classic cases are the Milnor Hauptvermutung counterexamples of non-combinatorial homeomorphisms of compact polyhedra, the proof by Novikov of the topological invariance of the rational Pontrjagin classes, and the structure theory of Kirby and Siebenmann for high-dimensional compact topological manifolds. The open manifolds arise geometrically as tame ends: in the applications it is necessary to close them. The obstruction theory for closing tame ends of open manifolds is also the obstruction theory for deciding if a finitely dominated space is homotopy equivalent to a finite C W complex.

Additive L-Theory

The cobordism groups of quadratic Poincar6 complexes in an additive category with involution A are identified with the Wall L-groups of quadratic forms and formations in A, generalizing earlier work for modules over a ring with involution by avoiding kernels and cokernels.

Surgery obstructions of fibre bundles

In a previous paper we obtained an algebraic description of the transfer maps p∗:Ln(Z[π1(B)]→Ln+d(Z[π1(e)]) induced in the Wall surgery obstruction groups by a fibration with the fibre F a d-dimensional Poincare complex. In this paper we define a Π1(B)-equivariant symmetric signature σ∗(F, ω)ϵLd(π1(b), Z) depending only on the fibre transport ω: Π1(B)→[F, F], and prove that the composite p∗p∗:Ln(Z[π1(B)])→Ln+d(Z[π1(b)] is the evaluation σ∗(F, ω)⊗? of the product ⊗:Ld(π1(B), Z⊗Ln(Z[π1(B)])→Ln+d(Z[π1(B)]). This is applied to prove vanishing results for the surgery transfer, such as p∗=0 if F=G is a compact connected d-dimensional Lie group which is not a torus, and is a G-principal bundle. An appendix relates this expression for p∗p∗ to the twisted signature formula of Atiyah, Lusztig and Meyer.

THE WHITEHEAD GROUP OF THE NOVIKOV RING

The Bass-Heller-Swan-Farrell-Hsiang-Siebenmann decomposition of the Whitehead group

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

K_1(A_{\rho}[z,z^{-1}])