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We are given an $R^4$ bundle $\xi$ over $S^4$, whose total space is E, and we know that the associated sphere bundle is the Hopf fibration $S^3 \rightarrow S^7 \rightarrow S^4$. How can we show that the total space E' of the associated disk bundle is $HP^2-$ an open 8-cell?

I know that the Thom space $T$ of $\xi$ is obtained from $E'$ by gluing a cone over $\partial E'\approx S^7$ and cell decomposition of $T$ is the same as that of $HP^2$. But I cannot combine these..

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1 Answer 1

Define an $\mathbb{H}$-bundle $$\mathbb{H} \hookrightarrow \mathbb{H}P^2 - \{[0:0:1]\} \xrightarrow{~\pi~} \mathbb{H}P^2,$$ $$\pi([u:v:w]) = [u:v] \in \mathbb{H}P^1$$ as follows. $\mathbb{H}P^1$ decomposes as the union of two disks $$D_1 = \{[u:1] : \|u\| \leq 1\},$$ $$D_2 = \{[1:v] : \|v\| \leq 1\}$$ glued along their common boundary via the map $[u:1] \mapsto [1:u^{-1}]$. Then take the local trivializations of our bundle to be $$\psi_1: D_1 \times \mathbb{H} \longrightarrow \pi^{-1}(D_1),$$ $$\psi_1([u:1],w) = [u:1:w],$$ $$\psi_2: D_2 \times \mathbb{H} \longrightarrow \pi^{-1}(D_2),$$ $$\psi_2([1:v],w) = [1:v:w].$$ Making the identifications $\mathbb{H} \cong \mathbb{R}^4$ and $\mathbb{H}P^1 \cong S^4$, this defines an $\mathbb{R}^4$-bundle over $S^4$. The transition function on the equatorial $S^3$ is $$\psi_2^{-1} \psi_1([u:1],w) = \psi_2^{-1}([u:1:w]) = \psi_2^{-1}([1:u^{-1}:u^{-1}w]) = ([1:v],vw).$$

Now from the total space $\mathbb{H}P^2 - \{[0:0:1]\}$ of our bundle, remove the open $8$-disk $$D = \{[u:v:1] : \|u\|^2 + \|v\|^2 < 1\}$$ centered at $[0:0:1]$. For every $[u:v] \in \mathbb{H}P^1$, this restricts the fiber over $[u:v]$ to $$\{[u:v:w] : \|w\|^2 \leq \|u\|^2 + \|v\|^2\} \cong D^4.$$ Therefore we have a disk bundle $$D^4 \hookrightarrow \mathbb{H}P^2 - D \xrightarrow{~\pi~} S^4$$ with transition function $$\psi_2^{-1} \psi_1([u:1], w) = ([1:v], vw) \tag{$\ast$}$$ on the equatorial $S^3$.

$S^3$-bundles over $S^4$ are classified by elements of $$\pi_3(\mathrm{SO}(4)) \cong \mathbb{Z} \oplus \mathbb{Z}.$$ An explicit isomorphism identifies the pair $(h,j) \in \mathbb{Z} \oplus \mathbb{Z}$ with the $S^3$-bundle over $S^4$ with transition function $$f_{hj}: S^3 \longrightarrow \mathrm{SO}(4),$$ $$f_{hj}(u) \cdot v = u^h v u^j$$ on the equatorial $S^3$, where here we consider $u \in S^3$ and $v \in \mathbb{R}^4$ as quaternions, i.e. the expression $u^h v u^j$ is understood as quaternion multiplication. From $(\ast)$, we see that the bundle $$D^4 \hookrightarrow \mathbb{H}P^2 - D \xrightarrow{~\pi~} S^4$$ has the transition function $f_{10}$. You can easily check that the Hopf bundle $$S^3 \hookrightarrow S^7 \longrightarrow S^4$$ has transition function $f_{10}$ as well, so by the above identification of $S^3$-bundles over $S^4$, its associated disk bundle must be isomorphic to $$D^4 \hookrightarrow \mathbb{H}P^2 - D \xrightarrow{~\pi~} S^4.$$ Therefore the total space of the disk bundle associated to the Hopf bundle over $S^4$ is $\mathbb{H}P^2$ with an open $8$-disk removed.

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