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F. T. Farrell

Publications and source records attributed to F. T. Farrell.

15 recordsLinked to original sources
arXiv2016

Teichmüller space of negatively curved metrics on Complex Hyperbolic Manifolds is not contractible

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists a closed smooth complex hyperbolic manifold $M$ with real dimension $n$ having non-trivial $π_1(\mathcal{T}^{<0}(M))$. $\mathcal{T}^{<0}(M)$ denotes the Teichmüller space of all negatively curved Riemannian metrics on $M$, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of $M$ that are homotopic to the identity.

math.GT↗
arXiv2009

Obstructions to Fibering a Manifold

Given a map f: M \to M of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S^1, these torsion obstructions are identified with the ones due to Farrell. We have changed the exposition according to the comments of the referee and corrected some typos. The paper will appear in Geometriae Dedicata.

math.GT↗
arXiv2006

The Teichmüller Space of Pinched Negatively Curved Metrics on a Hyperbolic Manifold is not Contractible

For a smooth manifold $M$ we define the Teichmüller space $\cT(M)$ of all Riemannian metrics on $M$ and the Teichmüller space $\cT^ε(M)$ of $ε$-pinched negatively curved metrics on $M$, where $0\leqε\leq\infty$. We prove that if $M$ is hyperbolic the natural inclusion $\cT^ε(M)\hookrightarrow\cT(M)$ is, in general, not homotopically trivial. In particular, $\cT^ε(M)$ is, in general, not contractible.

math.DG↗
arXiv2004

Finite automorphisms of negatively curved Poincare Duality groups

In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a $δ$-hyperbolic Poincare Duality group, then the fixed subgroup is a Poincare Duality group over Z/p. We also provide examples to show that the fixed subgroup might not even be a Duality group over Z.

math.GR↗
arXiv2004

EZ-structures and topological applications

We introduce the notion of an EZ-structure on a group. Delta-hyperbolic groups and CAT(0)-groups have EZ-structures. We show torsion-free groups having an EZ-structure automatically have an action by homeomorphisms on a closed (high-dimensional) ball, which is well-behaved away from a "bad limit set" in the boundary of the ball. We show that groups having such an action satisfy the Novikov conjecture. For torsion-free delta-hyperbolic groups $G$, we also give a lower bound for the homotopy groups $π_n(P(BG))$, where $P$ is the stable topological pseudo-isotopy functor.

math.GT↗
arXiv2004

Involutions of negatively curved groups with wild boundary behavior

We consider pairs (X,Y) where X is a compact, locally CAT(-1) space, and Y is a totally geodesic subspace. The inclusion induces an embedding of the boundaries at infinity of the universal covers; we focus on the case where these are spheres whose dimensions differ by 2. We show that if the embedding is tame, then it is unknotted. We give examples of pairs for which the embedding is knotted (and can be realized as the fixed point set of an involution). We also provide a criterion for knottedness of tame codimension two spheres in high dimensional (>5) spheres. Corresponding results for delta-hyperbolic groups are also discussed. Some corollaries give new results even in the Riemannian manifold case.

math.GT↗