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Suppose $\Omega\subset R^{n}$ is an open,convex and bounded set,$f:\Omega\to\mathbb{C}$ is a smooth map.

My question:

1)when $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$? Or in order to make $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$, whether there is some non-trivial restrictions on $f$?

What's more,if we do not need $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$,then comes the following:

2)How does $f$ affect $\pi_{1}(f(\Omega))$?

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Clearly not, the image could be a circle for instance (for $n=1$, say). – Torsten Ekedahl Jan 11 '11 at 10:59
f either injective or constant, perhaps? – Ketil Tveiten Jan 11 '11 at 12:23
This question is much too general. One could state a million conditions, but without knowing where this question comes from, they are almost certain to be useless. – Igor Rivin Jan 11 '11 at 16:02

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