# When is the union of embedded smooth manifolds a smooth manifold?

Suppose we have k embeddings of one single smooth manifold into one other, such that the intersections are manifolds,too. What are sufficient conditions, such that the union of those embeddings is a smooth submanifold in the image manifold?

To be more precise: Suppose we have a manifold M and a manifold N (both smooth). Moreover we have k-many embeddings $f_1: M \rightarrow N$ , $\ldots$, $f_k: M \rightarrow N$ such that the intersections of $f_j(M) \cap f_i(M) \neq \emptyset$ are in general not empty but it is required that $f_j(M) \cap f_i(M)$ is itself a submanifold of $N$. Is it possible at all that the union $\cup f_j(M)$ for all $1 \leq j \leq k$ is a submanifold of N?

Of course it is if all $f_j$'s are equal, but suppose some of them are not. Then in general the union is not a submanifold as we can see for example if we embedded $\mathbb{R}^2$ into $\mathbb{R}^3$ by $f_1(x_1,x_2):=(0,x_1,x_2)$ and $f_2(x_1,x_2):=(x_1,x_2,0)$ and $f_3(x_1,x_2):=(x_1,0,x_3)$. Then the intersections are lines and hence are manifolds by themself, but the union of the images of the $f_j$'s is not a manifold.

The question is: Are there conditions under which the union is a submanifold or not?

Of course one sufficient condition is that there is a $i$ such that $f_i(M)=\cup f_j(M)$ $i \neq j$. So the more interesting situation is, when we have $f_i(M)\neq \cup f_j(M)$ for all $i \neq j$.

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Do you want to assume your manifolds are closed (compact with empty boundary)? –  Mark Grant Oct 21 '11 at 7:04
I think there are two possible reasons for the union of the embedded sumbanifolds $M_k\subset N$ not being a submanifold of $N$: i) some intersection $M_k\cap M_j$ has lower dimension; ii) some $M_i$ has a (topological) boundary in $N$ that meets another $M_k$. –  Pietro Majer Oct 21 '11 at 7:19
This question lacks motivation and could be homework. It seems not suitable for MO to me. –  Benoît Kloeckner Oct 21 '11 at 9:42
As a motivation I guess that the spirit of the question is, to provide a criterion to make a submanifold patching together smaller pieces (of course under this respect closed submanifolds are a trivial case) –  Pietro Majer Oct 21 '11 at 11:47
I had a typo in my question... The intersections are manifolds but NOT images of the preimage manifold. In the situations I have in mind, they are of lower dimension. Moreover nothing like being compact or closed is assumed. I can't find anything about it and I think this is not trivial or a homework exercise at all. –  Mirco Oct 21 '11 at 14:03

For simplicity, I will assume that all manifolds are connected and closed.

An obvious sufficient condition is that the for each pair of embeddings, their images are either disjoint or equal.

To see that this is also necessary, suppose that the two embeddings $f\colon M \hookrightarrow N,\, f'\colon M' \hookrightarrow N$ of copies of $M$ intersect in a submanifold $Q$. Note that $Q = f^{-1}(M') \subseteq M$ can be viewed as a closed submanifold of $M$. This implies that either $Q=M=M'$, or $Q$ has dimension less than that of $M$. In this latter case, the union $f(M)\cup f'(M')$ will not be a submanifold (each point of $Q$ is a double-point singularity of the immersion $f\sqcup f'$).

The same argument extends to disconnected manifolds and shows that the images of each connected component must be pairwise equal or disjoint.

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The intersections are manifolds but NOT images of the preimage manifold. Sorry for that! –  Mirco Oct 21 '11 at 14:03

The union will usually not be a manifold. Here are three examples:

1) x (two intersecting lines)

2) 8 (two tangent circles)

3) The topologist's sine curve, the union of two disjoint embeddings of R.

Based on these, I'm guessing there's no local condition that guarantees the union is a manifold. You probably can't do much better than requiring the manifolds to be closed and disjoint.

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To expand on my comment, say for all $i$, $M_i$ is an $m$-dimensional submanifold of $N$. So in your question, $M_i = f_i(M)$. But the $M_i$'s need not be diffeomorphic for the answer to hold, below.

Further assume that $\overline{M_i} \cap M_j = \overline{M_i \cap M_j} \cap M_j$ for all pairs $i \neq j$, and $M_i \cap M_j$ is also an $m$-dimensional submanifold of $N$ for all $i \neq j$.

Then I claim $\cup_i M_i$ is an $m$-dimensional submanifold of $N$. The proof is fairly mechanical, basically the closure condition rules out the "topologist's sine curve" example Anton Lukayenko gives. It ensures that for any point in the union, a chart neighbourhood for an $M_i$ can be restricted to a chart neighbourhood of any overlapping $M_j$ because $M_j$ can't approach from a transverse direction. The condition that intersections all are $m$-dimensional rules out the figure-8 case, etc.

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Unfortunately the requirement $dim(M_i \cap M_j)=m$ for $dim(M_i)=m$ is not true in the simplicial setting –  Mirco Oct 21 '11 at 18:44
What do you mean? Certainly it can be true, unless you add some extra hypothesis that nullifies this possibility. For example, in $\mathbb R^2$ you can take the set $\{(x,y): x > -1, y=0 \}$ together with $\{(x,y): x < 1, y=0 \}$. –  Ryan Budney Oct 21 '11 at 21:09
Its worth noting that every closed manifold arises as such a union: just take a finite atlas. –  Mark Grant Oct 22 '11 at 7:30
Maybe its a manifold, iff the coproduct exists as a manifold. This is just an idea I had today and I have to go deeper into it. But maybe someone else knows more... –  Mirco Oct 23 '11 at 22:31