# Boundary of a manifold is a submanifold?

I was reading in the book Morse Theory and Floer Homology by Audin and Damian (translated in english) that the boundary of a manifold is not always a submanifold. I cannot see why that is true. Any explanations or an example of such a case would be great.

I am using the definitions of manifold and submanifold from Jeffrey Lee's Manifolds and Differential Geometry.

• Boundary has two different meanings. In differential geometry, $(0,1)$ is a manifold without boundary. (After all, it is the same manifold as $\mathbb R$...) – PseudoNeo Sep 8 '15 at 12:37
• I don't understand. Page 538, the authors write "The boundary, denoted by ∂V, is a submanifold of dimension n − 1." Where do they say otherwise? – PseudoNeo Sep 8 '15 at 12:41
• This should help you answer your question. – RayX Sep 8 '15 at 16:24
• @PseudoNeo pg100 - "It is however not completely true $\partial M$ is a submanifold of M." – u184 Sep 8 '15 at 20:15
• @RayX Thanks, that helped. So the boundary is indeed a submanifold. I am guessing that maybe the author wanted to be really careful for the construction that followed. – u184 Sep 8 '15 at 20:34

## 1 Answer

I don't have the book in front of me, but the authors might mean that the boundary (in the topological sense) of a submanifold need not be a submanifold. For example, an open square $(0,1)\times(0,1)$ is a submanifold of $\mathbb{R}^2$, but its boundary is not a smooth submanifold, as it has corners (you can easily get topological counterexamples, by for example removing $\{0\}\times(\frac 12,1)$ from the open square).