Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

My first question is whether $m$-th symmetric product of $\mathbb{C}^{n}$ is simply connected, where $n\geq 3$.

The second question is whether $Hilb^{m}(\mathbb{C}^{n})$ is simply connected, where $n\geq 3$.

If not, how about the case when $m\gg1$.

share|cite|improve this question
For the first one: it's closed under dilation, so contractible. –  Allen Knutson Jun 21 '13 at 3:17
If $Hilb^m(C^n)$ the Hilbert scheme of points? –  Mariano Suárez-Alvarez Jun 21 '13 at 3:34

1 Answer 1

Re your first question: If $\pi_1(X)=H_1(X,{\mathbb Z})=0$, then any $m$-th symmetric product of $X$ is simply connected; this is a special case of Theorem 1.1 in this paper by Kallel and Taamallah.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.