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David J. Wraith

Publications and source records attributed to David J. Wraith.

15 recordsLinked to original sources
arXiv2024

On the topology of the moduli space of positive scalar curvature concordances

Let $M$ be a manifold which admits a metric with positive scalar curvature (or a positive intermediate curvature in a suitable sense). We study the moduli space ${\mathscr{M}}^{{\mathsf{pos}}_*}_{\sqcup}(M\times I)_g$ of concordances of such metrics (with appropriate boundary conditions) which restrict to a given metric $g$ on $M \times \{0\} \cup\partial M \times I$. We show that $\pi_{4*}{\mathscr{M}}^{{\mathsf{pos}}_*}_{\sqcup}(M \times I)_g \otimes {\mathbb Q} \neq 0$ in a stable range provided $\dim M$ is even. We obtain analogous results when positive scalar curvature is replaced by $k$-positive Ricci curvature for $k \ge 2$.

math.DG↗
arXiv2023

Positive intermediate Ricci curvature on connected sums

We consider the problem of performing connected sums in the context of positive $k^{th}$ intermediate Ricci curvature. We show that such connected sums are possible if the manifolds involved possess `$k$-core metrics' for some $k$. Here, a $k$-core metric is a generalization of the notion of core metric introduced by Burdick for positive Ricci curvature. Further, we show that connected sums of linear sphere bundles over bases admitting such metrics admit positive $k^{th}$ intermediate Ricci curvature for $k$ in a particular range. This follows from a plumbing result we establish, which generalizes other recent plumbing results in the literature and is possibly of independent interest. As an example of a manifold admitting a $k$-core metric, we prove that $\mathbb{H} P^n$ admits a $(4n-3)$-core metric and that $\mathbb{O}P^2$ admits a $9$-core metric, and we show that in both cases these are optimal.

math.DG↗
arXiv2023

A generalization of the Perelman gluing theorem and applications

We extend a positive Ricci curvature gluing theorem of Perelman to a range of positive intermediate curvature conditions, ranging from positive scalar curvature up to (and including) positive sectional curvature. As an application of this, we demonstrate that the observer moduli space of metrics with positive intermediate Ricci curvatures can have non-trivial higher homotopy groups. Further applications include deriving a sufficient condition for the existence of a metric with positive intermediate Ricci curvature and totally geodesic boundary.

math.DG↗
arXiv2022

Positive Intermediate Ricci Curvature on Fibre Bundles

We prove a canonical variation-type result for submersion metrics with positive intermediate Ricci curvatures. This can then be used in conjunction with surgery techniques to establish the existence of metrics with positive intermediate Ricci curvatures on a wide range of examples which had previously only been known to admit positive Ricci curvature, such as highly connected manifolds and exotic spheres. Further, we extend results of the second author on the moduli space of metrics with positive Ricci curvature to positive intermediate Ricci curvatures.

math.DG↗
arXiv2022

Intermediate Ricci curvatures and Gromov's Betti number bound

We consider intermediate Ricci curvatures $Ric_k$ on a closed Riemannian manifold $M^n$. These interpolate between the Ricci curvature when $k=n-1$ and the sectional curvature when $k=1$. By establishing a surgery result for Riemannian metrics with $Ric_k>0$, we show that Gromov's upper Betti number bound for sectional curvature bounded below fails to hold for $Ric_k>0$ when $\lfloor n/2 \rfloor+2 \le k \le n-1$. This was previously known only in the case of positive Ricci curvature.

math.DG↗
arXiv2020

H-Space and Loop Space Structures for Intermediate Curvatures

For dimensions $n\geq 3$ and $k\in\{2, \cdots, n\}$, we show that the space of metrics of $k$-positive Ricci curvature on the sphere $S^{n}$ has the structure of an $H$-space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May that the path component of this space containing the round metric is weakly homotopy equivalent to an $n$-fold loop space.

math.DG↗
arXiv2018

On the topology of the space of Ricci-positive metrics

We show that the space $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n})$ of metrics with positive Ricci curvature on the manifold $W^{2n}_g := \sharp^g (S^n \times S^n)$ has nontrivial rational homology if $n \not \equiv 3 \pmod 4$ and $g$ are both sufficiently large. The same argument applies to $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n} \sharp N)$ provided that $N$ is spin and $W_g^{2n} \sharp N$ admits a Ricci positive metric.

math.AT↗
arXiv2017

Homotopy groups of the observer moduli space of Ricci positive metrics

The observer moduli space of Riemannian metrics is the quotient of the space $\mathcal{R}(M)$ of all Riemannian metrics on a manifold $M$ by the group of diffeomorphisms $\mathrm{Diff}_{x_0}(M)$ which fix both a basepoint $x_0$ and the tangent space at $x_0$. The group $\mathrm{Diff}_{x_0}(M)$ acts freely on $\mathcal{R}(M)$ providing $M$ is connected. This offers certain advantages over the classic moduli space, which is the quotient by the full diffeomorphism group. Results due to Botvinnik, Hanke, Schick and Walsh, and to Hanke, Schick and Steimle have demonstrated that the higher homotopy groups of the observer moduli space $\mathcal{M}_{x_0}^{s>0}(M)$ of positive scalar curvature metrics are, in many cases, non-trivial. The aim in the current paper is to establish similar results for the moduli space $\mathcal{M}_{x_0}^{\mathrm{Ric}>0}(M)$ of metrics with positive Ricci curvature. In particular we show that for a given $k$, there are infinite order elements in the homotopy group $\pi_{4k}\mathcal{M}_{x_0}^{\mathrm{Ric}>0}(S^n)$ provided the dimension $n$ is odd and sufficiently large. In establishing this we make use of a gluing result of Perelman. We provide full details of the proof of this gluing theorem, which we believe have not appeared before in the literature. We also extend this to a family gluing theorem for Ricci positive manifolds.

math.DG↗
arXiv2016

Path-component invariants for spaces of positive scalar curvature metrics

The Kreck-Stolz $s$-invariant is a classic path-component invariant for the space and moduli space of positive scalar curvature metrics. It is an absolute (as opposed to relative) invariant, but this strength comes at the expense of being defined only under restrictive topological conditions. The aim of this paper is to construct an analogous invariant for certain product manifolds on which the $s$-invariant is not defined.

math.DG↗
arXiv2016

Non-negative versus positive scalar curvature

We show that results about spaces or moduli spaces of positive scalar curvature metrics proved using index theory can typically be extended to non-negative scalar curvature metrics. We illustrate this by providing explicit generalizations of some classical results concerning moduli spaces of positive scalar curvature metrics.

math.DG↗
arXiv2015

Positive Ricci curvature on highly connected manifolds

For $k \ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k \equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $Σ^{4k-1}$ such that $M \sharp Σ$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.

math.DG↗
arXiv2011

On the curvature of G-manifolds with finitely many non-principal orbits

We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of examples to which these existence results apply.

math.DG↗
arXiv2011

On the topology of G-manifolds with finitely many non-principal orbits

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that the Lie group action must have odd cohomogeneity. We pay special attention to manifolds with one and two singular orbits, and construct some infinite families of examples. To illustrate the diversity within some of these families, we also investigate homotopy types.

math.DG↗