Suppose I have $M$ and $N$, two $k$-manifolds in $\mathbb{R}^n$. Is it true that $M\cup N$ is also a manifold? What is a sufficient condition for positive answer?
Is the union of two manifolds a manifold?
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differential-geometry
manifolds
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3I don't think there's really much to say besides that $M\cup N$ is a manifold iff near any $p\in M\cap N$, there's a neighborhood of $p$ in $M\cup N$ that looks like (an open subset of) $\mathbb{R}^k$. – 2012-04-29
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0related: http://mathoverflow.net/questions/78733/when-is-the-union-of-embedded-smooth-manifolds-a-smooth-manifold – 2017-01-28
1 Answers
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No. Take two lines in $\mathbb{R}^n$ which intersect only at the origin. Disjointness is sufficient, although not necessary, for a positive answer.
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1why two intersecting line is not a manifold? – 2012-04-29
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5a neighborhood of the point of intersection looks like $X$ which is not homeomorphic to $\mathbb{R^n}$ for any $n$. – 2012-04-29
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6(i mean looks topologically like the letter x). To see that that's not so, note that you can remove a point from the letter X to leave it with four connected components, a property that isn't true of $\mathbb{R}^n$ for any $n$. – 2012-04-29
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0@countinghaus : Topologically the union $\cup$ does not have a meaning unless you are talking about topological disjoint union which is defined even if the two spaces have elements in common since we can force them to be disjoint by $X\simeq X\times \{1\}$ where $1$ is some "tag" so saying that disjoint union of two lines intersecting at the origin is not a manifold is not true since $L_1\times \{1\}$ does not intersect $L_2\times \{2\}$.am i right? – 2012-08-27
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0I don't think the OP is asking about disjoint unions since he is considering both manifolds as subsets of $\mathbb{R}^n$ (so the honest union makes sense), but if he is asking about disjoint unions then yes, the answer to his question is that it's always true. – 2012-09-03