In the unit sphere \(S^{p+q-1} \subset \mathbb{R}^{p+q}\) let \(S^{p-1}\) and \(S^{q-1}\) be the subspheres consisting of points whose last \(q\) and first \(p\) coordinates are zero, respectively.
Show that \(S^{p+q-1} - S^{p-1}\) deformation retracts onto \(S^{q-1}\), and is in fact homeomorphic to \(S^{q-1} \times \mathbb{R}^p\).
Show that \(S^{p-1}\) and \(S^{q-1}\) are not the boundaries of any pair of disjointly embedded disks \(D^p\) and \(D^q\) in \(D^{p+q}\). [The preceding exercise may be useful.]
Solution
Write points of \(S^{p+q-1}\) as \((x,y)\in\mathbb R^p\oplus\mathbb R^q\). Then \[ S^{p-1}=\{(x,0):\|x\|=1\}, \qquad S^{q-1}=\{(0,y):\|y\|=1\}. \]
1. There is a homeomorphism\[ S^{p+q-1}-S^{p-1}\cong S^{q-1}\times\mathbb R^p. \]
On the complement one has \(y\ne0\). Define \[ \Phi(x,y)=\left(\frac y{\|y\|},\frac x{\|y\|}\right). \] Its inverse is \[ \Phi^{-1}(u,v)=\frac{(v,u)}{\sqrt{1+\|v\|^2}}. \] These formulas are continuous and inverse to one another.
2. Hence \(S^{p+q-1}-S^{p-1}\) deformation retracts onto the coordinate sphere \(S^{q-1}\).
Under the homeomorphism in 1, contract the \(\mathbb R^p\) factor linearly to \(0\). The subspace \(S^{q-1}\) corresponds exactly to \(S^{q-1}\times\{0\}\).
3. Suppose, for contradiction, that there are disjoint embedded disks\[ D^p,D^q\subset D^{p+q} \]with boundaries the two coordinate spheres \(S^{p-1}\) and \(S^{q-1}\).Then the class\[ [S^{q-1}]\in H_{q-1}(S^{p+q-1}-S^{p-1})\cong\mathbb Z \]is a generator.
By <1>1–<1>2 the coordinate \(S^{q-1}\) is a deformation retract of the complement, so its fundamental homology class generates \(H_{q-1}\).
4. By Exercise 3, the inclusion\[ S^{p+q-1}-S^{p-1}\hookrightarrow D^{p+q}-D^p \]induces an isomorphism on homology.
Apply Exercise 3 to the properly embedded pair \((D^p,S^{p-1})\subset(D^{p+q},S^{p+q-1})\).
5. But \([S^{q-1}]\) maps to zero in \(H_{q-1}(D^{p+q}-D^p)\), a contradiction.
The disk \(D^q\) is disjoint from \(D^p\) by assumption, so it is a singular \(q\)-chain in \(D^{p+q}-D^p\) whose boundary is \(S^{q-1}\). Thus the image of \([S^{q-1}]\) is zero. This contradicts the injectivity in 4.
Therefore the two coordinate spheres cannot bound disjoint embedded disks in \(D^{p+q}\).