arXiv subjects

Francis Thomas Farrell

Publications and source records attributed to Francis Thomas Farrell.

3 recordsLinked to original sources

arXiv2020

Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$-equivariant homology group of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on $V$ have nontrivial rational homotopy groups.

math.GT↗

arXiv2019

Negatively curved bundles in the Igusa stable range

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.

math.GT↗

arXiv2018

On negatively curved bundles with hyperbolic fibers outside the Igusa stable range

We prove that the Teichmüller space $\mathcal{T}^{<0}(M)$ of negatively curved metrics on a hyperbolic manifold $M$ has nontrivial $i$-th rational homotopy groups for some $i> \dim M$. Moreover, some elements of infinite order in $π_i B\mbox{Diff}(M)$ can be represented by bundles over $S^i$ with fiberwise negatively curved metrics.

math.GT↗