Let $M^4$ be a noncompact, contractible, smooth manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) each $U_i$ is homeomorphic to $\mathbb R^2 \times S^2$. Can we prove that $M$ is homeomorphic to $\mathbb R^4$?
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3$\begingroup$ This seems pretty likely; you need to show that M is simply connected at infinity. Since M is contractible, it should have vanishing $H_1$ at infinity; the hypothesis says that there are copies of $S^1 \times S^2$ far out at infinity so commutators would be trivial. But that's just a heuristic argument. $\endgroup$– Danny RubermanCommented Aug 2, 2021 at 14:40
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