Search

We classify the possible elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary and conclude that they are either polycyclic or solvable Baumslag- Solitar. Since these groups are good and satisfy the Farrell-Jones Conjecture, one concludes that such manifolds satisfy topological rigidity: a homotopy equivalence which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism. We classify the closed 3-manifolds which arise as the boundary of an compact aspherical 4-manifold with elementary amenable fundamental group, generalizing results of Freedman and Quinn in the cases of trivial and infinite cyclic fundamental groups. Moreover, two such 4-manifolds are homeomorphic if and only if their "enhanced" peripheral group systems are equivalent, and each such manifold is the boundary connected sum of a compact aspherical 4-manifold with prime boundary and a contractible 4-manifold., Comment: 34 pages

We undertake a systematic investigation of compact aspherical manifolds with boundary; motivated by the plethora of examples in the bounded case and by the beauty of the theory in the closed case. Our main theorems give a homological criterion for when a closed manifold, together with maps from the fundamental groups of its components to a fixed group, can be realized as the boundary of a compact aspherical manifold. This is done in two steps: we first produce a Poincar\'e pair and then apply surgery theory to obtain a manifold. We illustrate this in the case of abelian fundamental group. The results of this paper will be applied in a sequel where we classify compact aspherical 4-manifolds with elementary amenable fundamental group., Comment: 29 pages

We consider the problem of whether, for a given virtually torsionfree discrete group $\Gamma$, there exists a cocompact proper topological $\Gamma$-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that $\Gamma$ contains a normal torsionfree subgroup $\pi$ such that $\pi$ is hyperbolic and $\pi$ is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and $\Gamma/\pi$ is a finite cyclic group of odd order., Comment: 41 pages. The final, final version, to appear in the Transactions of the American Mathematical Society

We show that the additive category of chain complexes parametrized by a finite simplicial complex $K$ forms a category with chain duality. This fact, never fully proven in the original reference, is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincar\'e duality to global Poincar\'e duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on $K$-based chain complexes., Comment: 30 pages, 4 figures, final version

We compute the topological simple structure set of closed manifolds which occur as total spaces of flat bundles over lens spaces S^l/(Z/p) with fiber an n-dimensjional torus T^n for an odd prime p and l greater or equal to 3, provided that the induced Z/p-action on pi_1(T^n) = Z^n is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained from torsionfree and finite groups using amalgamated and HNN-extensions. We give a collection of classical surgery invariants such as splitting obstructions and rho-invariants which decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism., Comment: 40 pages, to appear in Communications on Pure and Applied Mathematics

This paper undertakes a study of the structure of the fibers of the Chevalley exponentiation maps $f_{(i_1,\dots ,i_d)}$. The fibers of these maps $f_{(i_1,\dots ,i_d)}$ encode the nonnegative real relations amongst exponentiated Chevalley generators. Our main theorems show that the fibers admit cell stratifications, that these cell stratifications have the same face posets as interior dual block complexes of subword complexes, and that these posets are contractible. We conjecture that each such fiber is a regular CW complex homeomorphic to the interior dual block complex of a subword complex. This conjecture is shown to have as a corollary a new proof of the Fomin-Shapiro Conjecture by way of general topological results regarding approximating maps by homeomorphisms., Comment: Major revision with 36 pages added to prior version. New version is 63 pages with several figures and examples added, as well as much more detail in the proofs, and some further results

In a recent paper Baker and Bowler introduced matroids over hyperfields, offering a common generalization of matroids, oriented matroids, and linear subspaces of based vector spaces. This paper introduces the notion of a topological hyperfield and explores the generalization of Grassmannians and realization spaces to this context, particularly in relating the (hyper)fields R and C to hyperfields arising in matroid theory and in tropical geometry., Comment: 31 pages. Final version

A long-standing conjecture of Farrell and Zdravkovska and independently S.~T.~Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions., Comment: 8 pages, to appear in the Journal of Differential Geometry. New version of Lemma 2.5: A manifold bounds if there is an involution on TM whose fixed bundle is full rank

The problem of equivariant rigidity is the $\Gamma$-homeomorphism classification of $\Gamma$-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of $\Gamma$. In other words, this is the classification of cocompact $E_{fin}\Gamma$-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., Comment: 21 pages, slightly modified title, added a last section for examples

The main theorem is that if K is a finite CW complex with finite fundamental group G and universal cover homotopy equivalent to a product of spheres X, then G acts smoothly and freely on X x S^n for any n greater than or equal to the dimension of X. If the G-action on the universal cover of K is homologically trivial then so is the action on X x S^n. Unlu and Yalcin recently showed that for every finite group G, there is a finite CW complex K with fundamental group G which acts homologicially trivially on the universal cover of K. Thus every finite group acts smoothly, freely, and homologically trivially on a product of spheres., Comment: 11 pages. Final version. To appear in the Proceedings of the American Mathematical Society

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions., Comment: To appear in the Journal of the AMS. 32 pages

We prove there is only one involution (up to conjugacy) on the n-torus which acts as $-\mathrm{Id}$ on the first homology group when $n$ is of the form $4k$, is of the form $4k+1$, or is less than $4$. In all other cases we prove there are infinitely many such involutions up to conjugacy, but each of them has exactly $2^n$ fixed points and is conjugate to a smooth involution. The key technical point is that we completely compute the equivariant structure set for the corresponding crystallographic group action on $\mathbb{R}^n$ in terms of the Cappell $\mathrm{UNil}$-groups arising from its infinite dihedral subgroups. We give a complete analysis of equivariant topological rigidity for this family of groups., Comment: 50 pages, to appear in Geometry & Topology

This is a survey of some of the work of Tom Farrell and Lowell Jones. This is the lead article of a special issue of the Pure and Applied Mathematics Quarterly. This issue is published in conjunction with the conference "Geometry,Topology, and their Interactions" held in Morelia, Mexico., Comment: 14 pages. Typos corrected. Final version

Let Gamma be a semidirect product of the form Z^n rtimes Z/p where p is prime and the Z/p-action on Z^n is free away from the origin. We will compute the topological K-theory of the real and complex group C*-algebra of Gamma and show that Gamma satisfies the unstable Gromov-Lawson-Rosenberg Conjecture. On the way we will analyze the (co-)homology and the topological K-theory of the classifying spaces BGamma and underbar{B}Gamma. The latter is the quotient of the induced Z/p-action on the torus T^n., Comment: 46 pages. Final version. Accepted for publication in the Journal of Noncommutative Geometry

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection with finite kernel onto a cyclic group. The second is that the Waldhausen Nil groups for a group which maps epimorphically onto the infinite dihedral group can be computed in terms of the Farrell-Bass Nil groups of the index two subgroup which maps surjectively to the infinite cyclic group., Comment: Accepted for publication by the Journal of Topology, 23 pages, proof of Lemma 4.1 simplified

This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature., Comment: 8 pages

This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group.

We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.

We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space.

Four-dimensional surgery is used to show that a two component link with Alexander polynomial one is topologically concordant to the Hopf link., Comment: To appear in the Mathematical Proceedings of the Cambridge Philosophicial Society

Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S^2 x S^2's. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a certain map from the second homology of the fundamental group with coefficients in Z/2 to the L-theory of the group is injective. This injectivity is implied by the Borel/Novikov conjecture for torsion-free groups, which is known for many groups. There are also results concerning the homotopy invariance of the Kirby-Siebenmann invariant. The method of proof is to use Poincare duality in Spin bordism to translate between Wall's classical surgery and Kreck's modified surgery., Comment: Re-TeXed. The references are now numbered; Geometry and topology of manifolds, 63-76, Fields Inst. Commun., 47, Amer. Math. Soc., Providence, RI, 2005

This paper computes the quadratic Witt groups (the Wall L-groups) of the polynomial ring Z[t] and the integral group ring of the infinite dihedral group, with various involutions. We show that some of these groups are infinite direct sums of cyclic groups of order 2 and 4. The techniques used are quadratic linking forms over Z[t] and Arf invariants., Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper29.abs.html

We use the notion of fixity for representations of finite groups to construct free and smooth actions on products of spheres. In particular we show that a finite p-group (for p>3) will act freely and smoothly on a product of two spheres if and only if it does not contain a rank 3 elementary abelian subgroup. We show that if G is a finite subgroup of U(n), acting freely on U(n)/U(k) for some k>0 and if (|G|,(n-1)!)=1, then the action propagates to a free and smooth action on a product of n-k spheres. A number of explicit examples are discussed., Comment: 22 pages

We introduce a new spectral sequence called the p-chain spectral sequence which converges to the (co-)homology of a contravariant C-space with coefficients in a covariant C-spectrum for a small category C. It is different from the corresponding Atiyah-Hirzebruch type spectral sequence. It can be used in combination with the Isomorphism Conjectures of Baum-Connes and Farrell-Jones to compute algebraic K- and L-groups of group rings and topological K-groups of reduced group C*-algebras., Comment: 36 pages

This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.

We prove the Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups, thereby giving necessary and sufficient conditions for a closed spin manifold of dimension greater than four with fundamental group cocompact Fuchsian to admit a metric of positive scalar curvature., Comment: 6 pages

We show that there is no triangulation of the infinite real Grassmannian of k-planes in R^\infty which is nicely situated with respect to the coordinate axes. In terms of matroid theory, this says there is no triangulation of the Grassmannian subdividing the matroid stratification. This is proved by an argument in projective geometry, considering a specific sequence of configurations of points in the plane., Comment: 11 pages

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid bundles to spherical quasifibrations. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. This shows the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes. The homotopy groups of this poset are related to the image of the J-homomorphism from stable homotopy theory.

A free action of the direct product of two copies of the symmetric group on 3 elements on the cartesian product of two copies of the 3-sphere is constructed. This nonlinear action is constructed using surgery. The action provides a counterexample to a conjecture of Lewis made in 1968.

This paper surveys some results and methods in topological transformation groups.