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## Finite codimension in Banach spaces

On 2009/11/27 at IHP on the occasion of the Katok 65 conference, Patrick BERNARD presented his notion of codimension in Banach spaces with applications to Mather measures and the transversality theorem.

B will denote the ambient Banach space. A large part of the theory extends to Fréchet spaces (things get tougher once C^1 smoothness is involved).

A Lipschitz graph in $B$ of codimension $k$ is a subset of the graph of a Lipschitz function $C \to T$ where $B=C\oplus T$ and $\dim(T)=k$. A rectifiable set of codimension $k$ is the image by the canonical projection $B\times \mathbb R^n\to B$ of a Lipschitz graph of codimension $k+n$ in $B\times\mathbb R^n$.  A subset of $B$ is of codimension $k$ if it is a countable union of rectifiable subsets of that codimension.

Theorem (Zajicek 2008). The complement of a subset of positive codimension in a Banach space is Baire generic (i.e., a dense $G_\delta$) and prevalent.

Codimension behaves as expected with respect to C^1 maps: it is invariant under C^1 diffeomorphisms, it decreases the Fredholm index of a C^1 map, it is preserved by preimage under submersions.

It is then strikingly straightforward to prove the following applications:

Theorem. The set of potentials with k+1 Mather measures has codimension k in C^p

Theorem. The set of maps from a m-dimensional manifold to $\mathbb R^n$ that fail to be injective immersions has codimension n-2k

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