$\mathcal{C}^{r}$ topologies

The $\mathcal{C}^{r}$ Whitney (or strong) topology is a topology assigned to the space $\mathcal{C}^{r}(M,N)$ of mappings from a $\mathcal{C}^{r}$ manifold $M$ to a $\mathcal{C}^{r}$ manifold $N$ having $r$ continuous derivatives . It gives a notion of proximity of two $\mathcal{C}^{r}$ mappings, and it allows us to speak of “robustness” of properties of a mapping. For example, the property of being an embedding is robust: if $f\colon M\to N$ is a $\mathcal{C}^{r}$ embedding, then there is a strong $\mathcal{C}^{r}$ neighborhood of $f$ in which any $\mathcal{C}^{r}$ mapping $g\colon M\to N$ is an embedding.

Given a locally finite atlas $\{(U_{i},\phi_{i}):i\in I\}$ and compact sets $K_{i}\subset U_{i}$ such that there are charts $\{(V_{i},\psi_{i}):i\in I\}$ of $N$ for which $f(K_{i})\subset V_{i}$ for all $i\in I$ , and given a sequence $\{\epsilon_{i}>0:i\in I\}$ , we define the basic neighborhood

$\mathcal{U}^{r}\left(f,\phi,\psi,\{K_{i}:i\in I\},\{\epsilon_{i}:i\in I\}\right)$

as the set of $C^{r}$ mappings $g\colon M\to N$ such that for all $i\in I$ we have $g(K_{i})\subset V_{i}$ and

$\sup_{x\in\phi_{i}(K_{i}),0\leq k\leq r}||D^{k}(\psi_{i}f\phi_{i}^{-1})(x)-D^{% k}(\psi_{i}g\phi_{i}^{-1})(x)||<\epsilon_{i}.$

That is, those maps $g$ that are close to $f$ and have their first $r$ derivatives close to the respective first $r$ -th derivatives of $f$ , in local coordinates. It can be checked that the set of all such neighborhoods forms a basis for a topology, which we call the Whitney or strong $\mathcal{C}^{r}$ topology of $\mathcal{C}^{r}(M,N)$ .

The weak $\mathcal{C}^{r}$ topology, or $\mathcal{C}^{r}$ compact-open topology, is defined in the same fashion but instead of choosing $\{(U_{i},\phi_{i}):i\in I\}$ to be a locally finite atlas for $M$ , we require it to be an arbitrary finite family of charts (possibly not covering $M$ ).

The space $\mathcal{C}^{r}(M,N)$ with the weak or strong topologies is denoted by $\mathcal{C}^{r}_{W}(M,N)$ and $\mathcal{C}^{r}_{S}(M,N)$ , respectively.

We have that $\mathcal{C}^{r}_{W}(M,N)$ is always metrizable (with a complete metric) and separable. On the other hand, $\mathcal{C}^{r}_{S}(M,N)$ is not even first countable (thus, not metrizable) when $M$ is not compact; however, it is a Baire space. When $M$ is compact, the weak and strong topologies coincide.

Title	$\mathcal{C}^{r}$ topologies
Canonical name	mathcalCrTopologies
Date of creation	2013-03-22 14:08:27
Last modified on	2013-03-22 14:08:27
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	5
Author	Koro (127)
Entry type	Definition
Classification	msc 57R12
Synonym	Whitney topology
Synonym	compact-open $\mathcal{C}^{r}$ topology
Synonym	weak $\mathcal{C}^{r}$ topology
Synonym	strong $\mathcak{C}^{r}$ topology
Related topic	ConjectureApproximationTheoremHoldsForWhitneyCrMNSpaces
Related topic	ApproximationTheoremAppliedToWhitneyCrMNSpaces

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$\mathcal{C}^{r}$ topologies