Fix a positive integer n.
We call local homeomorphism of R^n a function f:U--->R^n with: U open neighborhood of 0 \in R^n, f(0)=0, f injective continuous and open.
We identify two local homeomorphisms of R^n if and only if they coincide in a suitable neighborhood of 0.
We denote by LO the set of all equivalence classes of local homeomorphisms of R^n.
We call local diffeomorphism of R^n a local homeomorphism of R^n which is smooth and has a smooth inverse.
We identify two local homeomorphisms of R^n if and only if they coincide in a suitable neighborhood of 0.
We denote by LD the set of all equivalence classes of local diffeomorphisms of R^n. LO is a group with the compiosition of functions, and LD is a subgroup of LO.
Is LD a normal subgroup of LO? If yes then what is known about the quotient group LO/LD? References are very welcome.
No, it's never a normal subgroup, as can be seen by the one-dimensional case: Define f \in LD by f(x) = x^3 +x/4 and let g \in LH be equal to the identity outside (-1/2, 1/2) and not differentiable at points p_n \rightarrow 0 such that p_n \neq f(p_m) for all m, n (such a function can easily be constructed). Then g^{-1} f g is not differentiable in a neighborhood of 0. As the subgroup is not normal, the usual operation on cosets is not well-defined.
A little comment. In the definition of local homeomorphism f:U--->R^n there is no need to say that f is open. Openness follows from continuity and injectivity, according to the Brower invariance of domain principle. See the exposition written by Terry
Tao in his blog:
https://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/
Dear Tommaso, first of all, you have to check:
Typically, when you are looking at locally-defined maps, it is better to go the route of pseudogroups, rather than groups. The difference is in whether composition is defined at all. Indeed, if you are really thinking about coordinate transformations then they define pseudogroups, not groups.
May I add some references which might become useful?
1) "On the structure of local c>n hommeomorphims of euclidean n-space" Shlomo Sternberg,
https://www.jstor.org/stable/2372774?seq=1#page_scan_tab_contents
2) "The structure of the classical diffeomorphisms groups" Augustin Banyaga
http://link.springer.com/book/10.1007%2F978-1-4757-6800-8
Note, however, that these two references explain the Dynamical Systems perspective, and I don't know if this is exactly what you need.
Hoping I helped,
Stavros.
I think that this will help. I am looking for informations about the group of local homeomorphism of R?n because it is strictly linked with the group of linear isomorphisms of the tangent space to a n-dimensional topological manifold.
I am interested in any perspective.
Strictly speaking, a topological manifold won't have tangent spaces (at least, not tangent vector spaces), since tangency first appears in the differentiable case. For an n-dimensional differentiable manifold, the group of linear isomorphisms of any tangent space will be isomorphic to GL(n), which is finite-dimensional, as opposed to the group of local homeomorphisms of Rn, which is infinite-dimensional.
Typically, the linear isomorphisms of tangent spaces arise by differentiating local diffeomorphisms of manifolds, such as a local diffeomorphism of a manifold with itself, when one differentiates at a fixed point of the map.
I extended to topological manifolds the well known construction of the tangent bundle to smooth manifolds maintaining functoriality, the linear structure of the fibre, the linearity on the fibre of the differential of continuous functions.
I am trying to publish this result but, up to now, no journal (I tried 11 journals) accepted the paper because of its length (140 pages) although many reviewers have encouraged me by recognizing the value of my research.
The questions I posted are asked in order to understand the structure of the fibre and of the group of continuous and linear isomorphisms of the fibre. I have reason to believe that this will lead to interesting results which will made printable the research despite the length of the paper.
Dear Tommaso,
You can choose the basic ideas and publish them in the form of an article. And all of the research publish in the monograph. Journals are always limited in volume.
Sincerelly Yuri
It is likely that in the end I will do so. However I fear that, in this way, the work will be emptied and the resulting paper will be a list of small talks, so that publishers will reject it anyway.
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