Is the Connected Sum of Stein Manifolds Also Stein?

Question

I can’t seem to find the answer to the question in the title despite searching for a while. In the examples I can think of it seems true, though that isn’t many. Help is appreciated.

Answer 1

The connected sum of complex manifolds of dimension $\ge 2$ does not have a natural complex structure (sometimes, it does not have a complex structure at all!). A meaningful interpretation of your question then is:

Suppose that $X, Y$ are $n$-dimensional Stein manifolds. Is $X\# Y$ homeomorphic (diffeomorphic) to a Stein manifold?

The answer to this is negative whenever $n\ge 2$. More precisely, the connected sum of Stein manifolds is never even homotopy-equivalent to a Stein manifold, when $n\ge 2$.

The reason is that every $n$-dimensional Stein manifold $X$ is homotopy-equivalent to an $n$-dimensional CW complex, hence, $H_{2n-1}(X)=0$ unless $n\le 1$. But if $X, Y$ are connected open $2n$-dimensional manifold then the separating sphere in $X\# Y$ represents a nontrivial class in $H_{2n-1}(X\# Y)$.

On the other hand, the connected sum of Stein Riemann surfaces (with the natural complex structure) is again Stein. This is non-trivial and follows from the fact that a Riemann surface is Stein if and only if it is open (proven by Behnke and Stein).