Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold $X$ with an embedding $S^2\times S^2\setminus \Delta\to X$ on to a dense open subset.
Of course $X=S^2\times S^2$ is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to $S^2\times S^2$, with $h^2(X)=2$?h^2(X)=rank(H^2(X))=2$? And more generally, in four-dimensional, replace$S^2$by a compact Riemann surface and the same question. Even more generally, let$S^k\Sigma$be the$k$-th symmetric product of a compact Riemann surface, which is a$2k$-dimensional manifold. Consider$S^k\Sigma\times S^l\Sigma\setminus \Delta$, where$\Delta= { (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset }$. What kind of compatification of$S^k\Sigma\times S^l\Sigma\setminus \Delta$can we have? (How to describe the minimum condition in this case?) p.s. Many blow-up constructions give candidate compactifications, but they will increase the second betti number$h^2$. So I restrict this question to "minimal" compactifications. 3 Restricted to minimal compactifications Let$\Delta\subset S^2\times S^2$be the diagonal. Then$S^2\times S^2\setminus \Delta$is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold$X$with an embedding$S^2\times S^2\setminus \Delta\to X$on to a dense open subset. Of course$X=S^2\times S^2$is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to$S^2\times S^2$?, with$h^2(X)=2$? And more generally, in four-dimensional, replace$S^2$by a compact Riemann surface and the same question. Even more generally, let$S^k\Sigma$be the$k$-th symmetric product of a compact Riemann surface, which is a$2k$-dimensional manifold. Consider$S^k\Sigma\times S^l\Sigma\setminus \Delta$, where$\Delta= { (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset }$. What kind of compatification of$S^k\Sigma\times S^l\Sigma\setminus \Delta$can we have? (How to describe the minimum condition in this case?) p.s. Many blow-up constructions give candidate compactifications, but they will increase the second betti number$h^2$. So I restrict this question to "minimal" compactifications. 2 added 19 characters in body; added 19 characters in body Let$\Delta\subset S^2\times S^2$be the diagonal. Then$S^2\times S^2\setminus \Delta$is an open 4-foldfour dimensional manifold. By a compactification of it, I mean a closed 4-fold four dimensional manifold$X$with an embedding$S^2\times S^2\setminus \Delta\to X$on to a dense open subset. Of course$X=S^2\times S^2$is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to$S^2\times S^2$? And more generally, in four-dimensional, replace$S^2$by a compact Riemann surface and the same question. Even more generally, let$S^k\Sigma$be the$k$-th symmetric product of a compact Riemann surface, which is a$2k$-dimensional manifold. Consider$S^k\Sigma\times S^l\Sigma\setminus \Delta$, where$\Delta= { (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset }$. What kind of compatification of$S^k\Sigma\times S^l\Sigma\setminus \Delta\$ can we have?