If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?


Here a real analytic variety is the set of zeros of a real analytic function $f: \mathbb{R}^n \rightarrow \mathbb{R}$.


This is certainly true if $V$ is a compact real analytic manifold (without boundary). But what about varieties that are not manifolds? Answers in other settings (complex analytic varieties or real algebraic varieties) would be interesting too.



Contractible real analytic varieties
It is a consequence of Sullivan's work


*Sullivan, D.*, Combinatorial invariants of analytic spaces, Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969-1970, Lect. Notes Math. 192, 165-168 (1971). [ZBL0227.32005](https://zbmath.org/?q=an:0227.32005).


that every compact $k$-dimensional real-analytic subset $V\subset {\mathbb R}^n$ is a mod 2 pseudo-manifold: Every $k-1$-dimensional simplex in a triangulation of $V$ is contained in an even number of $k$-simplices. (This is immediate from the main result of Sullivan's paper about local structure of $V$ as a cone over a base of even Euler characteristic.) Now, take the sum of all $k$-simplices in the given triangulation of $V$. This will be a mod 2 cycle. Since there are no simplices of dimension $k+1$, this cycle is not a boundary. Hence, $H\_k(V, {\mathbb Z}\_2)\ne 0$. In particular, $V$ cannot be contractible. Note that this argument is pretty much the same as in the smooth case, once you have Sullivan's local result.


Sullivan's paper is freely available [here](https://www.math.stonybrook.edu/%7Edennis/publications/PDF/DS-pub-0007.pdf).


