Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: physics