Erik Kjaer Pedersen

Controlled algebra and the novikov conjectures for K- and L-theory

IN THIS paper we combine the methods of [S] with the continuously controlled algebra of [l] and the L-theory of additive categories with involution [22] to split assembly maps in Kand L-theory. Specifically, we prove the following theorems. Let I be a group with finite classifying space BT. Assume ET admits a compactification X (meaning X compact, and ET is an open dense subset) satisfying the following conditions, (denoting X ET by Y).

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”

Algebraic K-theory with continuous control at infinity

Let (E, Σ) be a pair of spaces consisting of a compact Hausdorff space Ē and a closed subspace Σ. Let U be an additive category. This paper introduces the category B(E, Σ; U of geometric modules over E with coefficients in U and with continuous control at infinity. One of the main results is to show that the functor that sends a CW complex X to the algebraic K-theory of B(cX, X; U) is a homology theory. Here cX is the closed cone on X and X is its base. The categories B(E, Σ; U) are generalizations of the categories C(Z; U) of geometric modules and bounded morphisms introduced by Pedersen and Weibel [8]. Here (Z, ϱ) is a complete metric space. If X is a finite CW complex and O(X) is the metric space open cone on X considered in [9], then there is an inclusion of categories C(O(X); U)→B(cX, X; U). A second main result is that this inclusion induces an isomorphism on K-theory. One advantage of the present approach is that B(E, Σ; U) depends only on the topology of (E, Σ) and not on any metric properties. This should make application of these ideas to problems involving stratified spaces, for example, more direct and natural.

Bounded surgery and dihedral group actions on spheres

If a finite group G acts freely and simplicially on a complex homotopy equivalent to a sphere S, then G has periodic Tate cohomology: H (G; Z) ∼= H (G; Z) for all i > 0. Swan proved in [26] that this condition was also sufficient. For free topological actions on S itself, the first additional restriction is: Theorem. [19] A finite dihedral group does not act freely and topologically on S. Milnor’s argument used the compactness of S as well as the manifold structure. In fact, for dihedral groups with periodic cohomology, i. e. of order 2n where n is odd we have,

Finite loop spaces are manifolds

One of the motivating questions for surgery theory was whether every finite H:space is homotopy equivalent to a Lie group. This question was answered in the negative by Hilton and Roitberg s discovery of some counterexamples [18]. However, the problem remained whether every finite H-space is homotopy equivalent to a closed, smooth manifold. This question is still open, but in case the H-space admits a classifying space we have the following theorem.

A finite loop space not rationally equivalent to a compact Lie group

We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5.

Stability in controlled L-theory

We prove a squeezing/stability theorem for delta-epsilon controlled L-groups when the control map is a fibration on a finite polyhedron. A relation with boundedly-controlled L-groups is also discussed.

Topological equivalence of linear representations for cyclic groups: II

Abstract In the two parts of this paper we prove that the Reidemeister torsion invariants determine topological equivalence of G -representations, for G  a finite cyclic group.

Universal geometric examples for transfer maps in algebraic K- and L-theory

In this paper we will be concerned with various geometrically defined transfer maps in algebraic K-theory. In [3], Ehrlich shows that given a fibration F i −→ E p −→ B with finitely dominated fiber and base, there is a homomorphism p∗ : K̃0(Zπ1B) → K̃0(Zπ1E) which is related to the finiteness obstruction of E, σ(E) by the formula: σ(E) = p∗(σ(B))+ i∗(σ(F ) · χ(B), where χ(B) denotes the Euler characteristic of B. Similarly, given a PL bundle Anderson [2] defines a homomorphism p∗ : W̃h(Zπ1B) → W̃h(Zπ1E) which, given a fiber homotopy equivalence F ′ f // F

A Survey of Bounded Surgery Theory and Applications

We begin by attempting to answer the question: What is bounded topology and why do people study it?