The observer moduli space of Riemannian metrics is the quotient of the
space of all Riemannian
metrics on a manifold
by the
group of diffeomorphisms
which fix both a basepoint
and the tangent space at
.
The group
acts freely on
provided that
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 Hanke, Schick and Steimle have
demonstrated that the higher homotopy groups of the observer moduli space
of
positive scalar curvature metrics are, in many cases, nontrivial. The aim in
the current paper is to establish similar results for the moduli space
of
metrics with positive Ricci curvature. In particular we show that for a given
,
there are infinite-order elements in the homotopy group
provided the
dimension
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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