TITLE: Manifolds with two coordinate charts
QUESTION [20 upvotes]: What is an early reference for the fact that if a compact, connected $n$-manifold $M$ is covered by two open sets homeomorphic to $\mathbb{R}^n$ then $M$ is homeomorphic to $S^n$?
And is it true that if $M$ is a compact, connected $n$-manifold with boundary, and if $M$ is covered by two open sets homeomorphic to $\lbrace(x_1,\ldots,x_n) \in \mathbb{R}^n  |  x_n \ge 0\rbrace$, then $M$ is a closed ball?

REPLY [21 votes]: I'll only discuss the first question (EDIT : Actually, I address the second question at the end).  As Agol pointed out in the comments, for $n \geq 5$ this is an easy consequence of Newman's 1966 proof of the Poincare conjecture in the topological category.
I don't know if it was explicitly stated earlier than this.  However, it can easily be derived from the main result of the paper
MR0126835 (23 #A4129)
Brown, Morton
The monotone union of open n-cells is an open n-cell. 
Proc. Amer. Math. Soc. 12 1961 812–814. 
In fact, this works in all dimensions (including $3$ and $4$).  
Brown's theorem is as follows.  Assume that $M$ is a topological $n$-manifold and that for all compact $K \subset M$, there exists some open set $U \subset M$ with $K \subset U$ and $U \cong \mathbb{R}^n$.  Then $M \cong \mathbb{R}^n$.  Brown's proof is clever, but completely elementary.
To get the desired result from this, assume that $X = U_1 \cup U_2$ with $U_i \cong \mathbb{R}^n$ and that $X$ is compact.  Let $\phi : \mathbb{R}^n \rightarrow U_1$ be a homeomorphism.  It is enough to prove that $X \setminus \{\phi(0)\} \cong \mathbb{R}^n$.  We will do this with Brown's theorem.  Consider a compact set $K \subset X \setminus \{\phi(0)\}$.  To verify Brown's criteria, it is enough to construct a homeomorphism $\psi : X \setminus \{\phi(0)\} \rightarrow X \setminus \{\phi(0)\}$ such that $\psi(K) \subset U_2$.     
For $r>0$, let $B(r) \subset \mathbb{R}^n$ be the ball of radius $r$.  The set $U_1 \setminus U_2$ is compact, so there exists some $R>0$ such that $U_1 \setminus \phi(B(R)) \subset U_2$.  Also, there exists some $\epsilon > 0$ such that $K \cap \phi(B(\epsilon)) = \emptyset$.  It is easy to construct a homeomorphism $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $f(B(\epsilon)) = B(2R)$ and $f(0)=0$ and $f|_{\mathbb{R}^n \setminus B(3R)} = \text{id}$.  We can therefore define a homeomorphism $\psi : X \setminus \{\phi(0)\} \rightarrow X \setminus \{\phi(0)\}$ by $\psi(p) = \phi \circ f \circ \phi^{-1}(p)$ for $p \in U_1 \setminus \{\phi(0)\}$ and $\psi(p) = p$ for $p \notin U_1$.  Clearly $\psi(K) \subset U_2$.

EDIT : Lee suggested that this might be able to address his second question too.  I thought a bit about it, and I believe that it can.  The key is the following "relative" version of Brown's theorem, which can be proven exactly like Brown's theorem.
Theorem : Let $(M,N)$ be a pair consisting of a topological $n$-manifold $M$ and a closed submanifold $N \subset M$.  Assume that for all compact $K \subset M$, there exists some open set $U \subset M$ such that $K \subset U$ and such that the pair $(U,U \cap N)$ is homeomorphic to the pair $(\mathbb{R}^n,\mathbb{R}^{n-1})$ (the second embedded in the standard way).  Then $(M,N) \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$.
To apply this, assume that $X$ is a compact manifold with boundary and that $X = U_1 \cup U_2$ with $(U_i,\partial U_i) \cong (\mathbb{R}^n_{\geq 0},\mathbb{R}^{n-1})$.  Double $X$ to get a closed manifold $Y$, and let $Y' \subset Y$ be the image of the boundary of $X$.  The open sets $U_i$ double to give an open cover $Y = V_1 \cup V_2$.  Letting $V_i' = V_i \cap Y'$, we have $(V_i,V_i') \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$.  Let $(M,M')$ be the result of deleting the image of $0$ in $(V_1,V_1')$.  It is enough to prove that $(M,M') \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$, and this can be proven just like above.

Of course, Agol answered the second question first -- it follows from the topological Schonfleiss theorem applied to the double, which was proven by Brown in
MR0117695 (22 #8470b) Reviewed
Brown, Morton
A proof of the generalized Schoenflies theorem.
Bull. Amer. Math. Soc. 66 1960 74–76.
54.00 (57.00)
Mazur had earlier proven a weaker result.  This requires the sphere to be bicollared, but this holds.  Indeed, from the assumptions the sphere is locally bicollared, and Brown proved in
MR0133812 (24 #A3637)
Brown, Morton
Locally flat imbeddings of topological manifolds.
Ann. of Math. (2) 75 1962 331–341. 
that this implies that the sphere is bicollared.  See 
MR0267588 (42 #2490)
Connelly, Robert
A new proof of Brown's collaring theorem.
Proc. Amer. Math. Soc. 27 1971 180–182. 
for a super-easy proof of Brown's collaring theorem.