Basic examples topological manifolds with boundary

Question

I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space $M$ such that

• $M$ is connected
• $M$ is Hausdorff
• $M$ is locally homeomorphic to $\mathbb{R}^n_+$

I have to show that

• $\mathring{B^n}:=\{x\in\mathbb{R^n}:\sum_i x_i^2<1\}$ is a topological manifold with boundary of dimension $n$

The first two conditions are trivial to check. Then let's consider $\Phi:\mathring{B^n}\to\mathbb{R}^n_+$ defined as $\Phi(x_1,\dots,x_n)=(x_1,\dots,e^{x_n})$ which is an homeomorphism.

• $B^n:=\{x\in\mathbb{R^n}:\sum_i x_i^2\le 1\}$ is a topological manifold with boundary of dimension $n$

As before, it's sufficient to check the last condition. Let's consider

$$ U_1=\{(x_1,\dots,x_n)\in B^n:x_n>-\frac{1}{2}\}\\ U_2=\{(x_1,\dots,x_n)\in B^n:x_n<\frac{1}{2}\} $$ and write $B^n=U_1\cup U_2$. Then I show that $U_i$ are homeomorphic to open subsets of $\mathbb{R}^n_+$ via the maps $$ \phi_1:U_1\to\mathbb{R}^n_+\\ (x_1,\dots,x_n)\mapsto (x_1,\dots,1-x_n) $$ and $$ \phi_2:U_2\to\mathbb{R}^n_+\\ (x_1,\dots,x_n)\mapsto (x_1,\dots,x_n+1) $$

Is it okay?

Answer 1

You didn't say what $\mathbb R^n_+$ is supposed to represent, but from the context, I assume it means $\{(x_1,\dots,x_n): x_n\ge 0\}$.

Your argument for $\mathring{B^n}$ is fine, except for one small mistake: You wrote, "Then let's consider $\Phi:\mathring{B^n}\to\mathbb{R}^n_+$ defined as $\Phi(x_1,\dots,x_n)=(x_1,\dots,e^{x_n})$ which is an homeomorphism." It's not a homeomorphism, because it's not surjective onto $\mathbb R^n_+$; but it's a homeomorphism onto an open subset of $\mathbb{R}^n_+$, which is good enough.

But your argument for $B^n$ has a number of problems. Here are the most important ones:

• Your two subsets $U_1$ and $U_2$ do not cover all of $B^n$.
• The images of $\phi_1$ and $\phi_2$ are not open subsets of $\mathbb R^n_+$. For example, $\phi_1(U_1)$ looks like a "bowl full of soup" sitting on the $x^n=0$ hyperplane. More formally, the point $(0,\dots,0) = \phi_1(0,\dots,0,1)$ is in the image of $\phi_1$, but any (relative) neighborhood of the origin in $\mathbb R^n_+$ would have to contain points of the form $(\varepsilon,0,\dots,0)$ for all sufficiently small $\varepsilon$, none of which are in the image of $\phi_1$.

To construct a homeomorphism from an open subset of $B^n$ to an open subset of $\mathbb R^n_+$, you need to find a map that "straightens out" the boundary sphere and maps it into the $x^n=0$ hyperplane. Stereographic projection might be useful for this.