Space Of Diffeomorphisms Research Articles

The Homotopy Type of the Space of Diffeomorphisms. II

The Homotopy Type of the Space of Diffeomorphisms. II

The homotopy type of the space of diffeomorphisms. I

A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result Diff ( D n , ∂ ) ≃ Ω n + 1 ( Top n / O n ) {\operatorname {Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text {Top}_n}/{O_n}) is obtained. The main technique is fibrewise smoothing.

The homotopy type of the space of diffeomorphisms. II

The result (proved in Part I) that Diff ( D n , ∂ ) ≃ Ω n + 1 ( PL n / O n ) {\operatorname {Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text {PL}_n}/{O_n}) is used to compute some new homotopy of Diff ( D n , ∂ D n ) {\operatorname {Diff}}({D^n},\partial {D^n}) . The relation between smooth and PL pseudo-isotopy is explored. Known and new results on the homotopy of PL n {\text {PL}_n} are summarized.

Diffeomorphisms of the 2-sphere

The analogue of Theorem A for the topological case was proved by H. Kneser [2]. The problem in his case seems to be of a different nature from the differentiable case. J. Munkres [3] has proved that Q is arcwise connected. Conversations with R. Palais have been helpful in the preparation of this paper. Let I2 be the square in the Euclidean plane E2 with coordinate (t, x) such that (t, x) E12 if O r> 1, and that all function spaces considered possess the Cr topology. We further assume that all diffeomorphisms are C. Let I1CI2 denote the subset { (t, x) EI21 t= 1 }, dfp be the differential of a diffeomorphism f at p, and uo be the vector (1, 0) in E2 considered as its own tangent vector space. Then denote by g the space of diffeomorphisms of 12 onto I2 such that if fEG, then (a) f = on some neighborhood of 12I,, and (b) dfp(uo) = uo for all p in some neighborhood of I,.

Stochastic metamorphosis in imaging science

In the pattern matching approach to imaging science, the process of \emph{metamorphosis} in template matching with dynamical templates was introduced in \cite{ty05b}. In \cite{HoTrYo2009} the metamorphosis equations of \cite{ty05b} were recast into the Euler-Poincare variational framework of \cite{HoMaRa1998} and shown to contain the equations for a perfect complex fluid \cite{Holm2002}. This result related the data structure underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids \cite{GBHR2013}. In particular, it cast the concept of Lagrangian paths in imaging science as deterministically evolving curves in the space of diffeomorphisms acting on image data structure, expressed in Eulerian space. (In contrast, the landmarks in the standard LDDMM approach are Lagrangian.) For the sake of introducing an Eulerian uncertainty quantification approach in imaging science, we extend the method of metamorphosis to apply to image matching along \emph{stochastically} evolving time dependent curves on the space of diffeomorphisms. The approach will be guided by recent progress in developing stochastic Lie transport models for uncertainty quantification in fluid dynamics in \cite{holm2015variational,CrFlHo2017}.