# Analytic stuctures on $\mathbb R^n$ and the nilpotent ideal of supermanifolds

I have two questions which are somewhat related:

(a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that this does not happen for any $\mathbb R^n$, $n\neq 4$. In other words, every smooth structure on $\mathbb R^n$, $n\neq 4$, is diffeomorphic to the standard structure. (Whose result is it?!)

MY QUESTION: What about analytic structures on $\mathbb R^n$? or complex-analytic structures on $\mathbb C^n$? Have these questions been answered?

(b) Let $(M,O_M)$ be a smooth (respectively, analytic) supermanifold (in the sense of Berezin, Kostant, Leites, Manin, etc.), that is, $M$ is a manifold and $O_M$ is a sheaf of $\mathbb Z_2$-graded algebras which is locally isomorphic to $C^\infty(\mathbb R^m)\otimes\Lambda \mathbb R^n$ (respectively, $C^\omega(\mathbb R^m)\otimes\Lambda \mathbb R^n$).

MY QUESTION: In his book "Gauge Field Theory and Complex Geometry", Manin defines the sheaf of ideals $J_M\subseteq O_M$ by $J_M=O_{M,1}^2+O_{M,1}$ and calls it the the "ideal generated by odd elements" (see $\S4.1.3$, page 182). Then he claims that for supermanifolds, $J_M$ is equal to the sheaf of ideals of nilpotent elements. I can prove (using partition of unity) that this statement is correct for smooth supermanifolds, but not for analytic supermanifolds. I have seen this statement in various places, but without a proper explanation.

I am not even sure why $O_{M,1}^2+O_{M,1}$ is a sheaf! Of course one can consider its sheafification, but I am not convinced that this is what they are doing, and in any case I don't have a counterexample.

• Michael Freedman found the first exotic $\Bbb R^4$ in 1982. As far as I know Milnor never did any work on exotic $\Bbb R^4$'s. – Henry T. Horton Oct 25 '13 at 1:15
• Thanks. Corrected. Sorry, I am by no means an expert in differential topology. – Valerie Oct 25 '13 at 2:00
• You might split this into 2 questions, since (a) and (b) are not very closely related. – Ben McKay Oct 25 '13 at 6:25
• (b): Since it is a sheaf, you only have to work locally, looking at nilpotent elements defined in an open set, so you can work with a ball in Euclidean space, if you like. This should give global supermanifold coordinates. – Ben McKay Oct 25 '13 at 6:27