High-Dimensional Manifold Topology

Preface.

Contents.

We prove a foliated control theorem for automorphisms of geometric modules. This is the analogue of a result for h-cobordism from [3].

We study a behavior of the conformal Laplacian operator 𝕃g on a manifold with tame conical singularities: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator 𝕃g on such manifolds. We describe the asymptotic of a general solution of the equation 𝕃gu = Qu^α with near each singular point. In particular, we derive the asymptotic of a Yamabe metric near such singularity.

We use the Fredholm resp. the unbounded picture for Kasparov KK-theory to define classifying spaces for the K-homology K_*(A) of a ℤ/2-graded σ-unital C*-algebra A. The classifying spaces emerging from the unbounded picture are used to define corresponding orthogonal spectra K(A) satisfying π_*K(A) ≅ K_*(A).

We construct 4k–dimensional generalized manifolds, k > 1, which have no resolutions. The construction proceeds as in a paper of Bryant, Ferry, Mio and Weinberger (see [1]) but does not use their controlled (∈, δ)–surgery sequence. The controlled surgery sequence is believed to be true. Recently, Pedersen, Quinn and Ranicki have given a proof of this sequence in the case of trivial local fundamental groups (see [4]).

In this work we develop the cellular equivariant homology functor and apply it to prove the equivariant Euler-Poincaré formula and the equivariant Lefschetz theorem.

We construct two classes of examples of a virtually torsion-free group G acting properly and cocompactly on a contractible manifold X. In the first class of examples the universal space for proper actions, , has no model with finitely many orbits of cells. The reason is that the centralizers of certain finite subgroups of G will not have finite-type classifying spaces. In the second class of examples X is a CAT(0) manifold upon which G acts by isometries. It follows that X is a model for . In these examples the fixed point sets of certain finite subgroups of G are not manifolds and the centralizers of these subgroups are not virtual Poincaré duality groups.

We show that, in contrast to the situation for the standard complex on which a right angled Coxeter group W acts, there are cocompact W-actions on CAT(0) complexes such that the local topology of the complex is distinctly different from the end topology of W.

Gromov and Lawson conjectured in [GL2] that a closed spin manifold M of dimension n with fundamental group π admits a positive scalar curvature metric if and only if an associated element in KO_n(Bπ) vanishes. In this note we present counter examples to the ‘if’ part of this conjecture for groups π which are torsion free and whose classifying space is a manifold with negative curvature (in the Alexandrov sense).

Let be a unital C*-algebra. We show that the set of regular unbounded -Fredholm operators respectively the set of self-adjoint regular unbounded -Fredholm operators on the standard Hilbert--module l²(A) equipped with the gap topology represent the functors which associate to a compact space X the groups and .

The Isomorphism Conjecture for stable pseudoisotopy theory formulates a simple recipe for constructing the Ω-spectrum of stable pseudoistopies on a connected manifold X from the collection of all Ω-spectra , where H denotes any virtually cyclic subgroup of π₁(X) and X_H denotes the covering space for X having H for fundamental group. In this paper some strong supporting evidence for the truth of this conjecture is developed, under the hypothesis that π₁(X) must act properly discontinuously by isometries on an A-regular complete Riemannian manifold which has non-positive sectional curvature values everywhere.

Let R denote a commutative principle ideal domain (a PID). Let (C_*, ∂_*) denote a chain complex of finitely generated torsion modules over R. We give a partial classification of such chain complexes: each such chain complex is isomorphic to a direct sum of a “minimal core” chain subcomplex with some “elemental” chain subcomplexes of (C_*, ∂_*); the isomorphism types of these summands is an invariant of the isomorphism type of (C_*, ∂_*). Examples of such chain complexes come from cellular actions of finite cyclic groups on finite CW complexes: in these examples the “minimal core” is a topological invariant of the group action.

Let Γ be a virtually cyclic group, and R be the ring of integers in an algebraic number field. We prove that K_–i(RΓ) = 0 for i > 1 and K₋₁(RΓ) is determined by the finite subgroups of Γ. We also prove similar results by replacing R with a finite field.

Let G = ℤ/2ℤ ≀ ℤ be the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1 – x is invertible, with x a generator of ℤ ⊂ G. Then R is not flat over kG. If k = ℂ, this applies in particular to the algebra of unbounded operators affiliated to the group von Neumann algebra of G.

We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us.

Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f: M → M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data. We prove the equivariant Lefschetz fixed point theorem, which says that these two classes agree. As a special case, we prove an equivariant Poincaré-Hopf Theorem, computing the universal equivariant Euler characteristic in terms of the zeros of an equivariant vector field, and also obtain an orbifold Lefschetz fixed point theorem. Finally, we prove a realization theorem for universal equivariant Euler characteristics.

For a normal covering over a closed oriented topological manifold we give a proof of the L²-signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the -version of the Baum-Connes conjecture imply the L²-signature theorem for a normal covering over a Poincaré space, provided that the group of deck transformations is torsion-free.

We discuss the various possible definitions of L²-signatures (using the signature operator, using the cap product of differential forms, using a cap product in cellular L²-cohomology, …) in this situation, and prove that they all coincide.

No abstract received.

The purpose of this paper is to discuss the four-periodicity of the topological surgery exact sequence from the point of view of controlled surgery.

We provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. Stability is a key ingredient in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Weinberger, but no proof has been available. The development given here is based on work of M. Yamasaki.

For proper actions of discrete groups topological K-theory K^*(X) can be defined using equivariant vector-bundles (see [5] along the lines of which we follow closely). We show that the analogous statement for proper smooth actions (i.e. actions with compact open isotropy groups) of totally disconnected groups isn’t true in general. An explicit counterexample will be given where excision doesn’t hold. For groups which are an inverse limit of discrete groups, the definition carries through and the Chern character of [4] can be extended to these groups.

We study power series over the group ring ℂF of a free group F. We prove that the von Neumann trace maps rational power series over ℂF to algebraic power series. Using the Riemann-Stieltjes formula, we deduce the rationality and positivity of Novikov-Shubin invariants of matrices over ℂF.

We give a detailed account of the Novikov complex corresponding to a closed 1-form ω on a closed connected smooth manifold M. Furthermore we deduce the simple chain homotopy type of this complex using various geometrically defined chain homotopy equivalences and show how they are related to another.

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