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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 29th 2011

had need for a stub for local diffeomorphism

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 2nd 2015

Suppose a smooth function $p \colon X \to \mathbb{R}^n$ from a diffeological space $X$ to Cartesian space induces at each point an isomorphism on tangent vectors as well as on all higher jets.

Then what sensible extra conditions does it take to conclude that $p$ is in fact a local diffeomorphism, i.e. restricts to a diffeomorphism around an open neighbourhood of each point?

Here I mean tangents and jets defined by equivalence classes of smooth maps into $X$.

• CommentRowNumber3.
• CommentAuthorigor
• CommentTimeMay 3rd 2015

Unless I’m missing something, but your condition implies that the Jacobian map $T p \colon TX \to T \mathbb{R}^n$ is full rank and even invertible. The inverse function theorem then guarantees that $p$ is a local diffeomorphism about any point $x\in X$ that has an $n$-manifold neighborhood. Is your question then about $X$’s that at some points fail to be $n$-manifold? But then, it seems to me, that essentially by definition there cannot be a local diffeomorphism from any neighborhood of such a point into $\mathbb{R}^n$.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 5th 2015

If you only know that $X$ is a diffeological space and that it has a map $X \to \mathbb{R}^n$ which is an iso on all tangents, what else does it need (if anything) to conclude that $X$ is a manifold sitting by a local diffeomorphism over $\mathbb{R}^n$?

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeMay 5th 2015
• (edited May 5th 2015)

What happened with moneomorphism and epiomorphism? I know that the terms are used rarely, especially outside of Eastern Europe, and especially the second term, but I remember writing about that terminology in nlab and this seemingly completely vanished. nLab search and google show no hits at nLab about those. Did I dream about writing it ?

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