The Milnor sphere and disk bundles $M_{h,j}$ and $W_{h,j}$

Let ${\xi }_{h,j}\to S^4$ be the oriented rank-four quaternionic clutching bundle of Quaternionic clutching bundles ${\xi }_{h,j}$ over $S^4$, with its Euclidean metric coming from the quaternionic norm $N$ on each fibre. Give ${\xi }_{h,j}$ that metric; it is preserved by the clutching maps and hence descends to a smooth bundle metric. Define $W_{h,j}:=D({\xi }_{h,j})=\{v\in {\xi }_{h,j}:\parallel v\parallel \le 1\},M_{h,j}:=S({\xi }_{h,j})=\{v\in {\xi }_{h,j}:\parallel v\parallel =1\}=\mathrm{\partial }{W}_{h,j},$ the closed disk bundle and the unit sphere bundle of Disk bundle, sphere bundle, and Thom space: the differential topology interface. Then $W_{h,j}$ is a compact oriented smooth eight-manifold with boundary, and $M_{h,j}=\mathrm{\partial }{W}_{h,j}$ is a closed oriented smooth seven-manifold, the total space of a smooth $S^3$-bundle $S^3⟶M_{h,j}⟶S^4\mathrm{.}$ The orientation of $W_{h,j}$ is the one for which the base orientation of $S^4$ followed by the fibre orientation of ${\xi }_{h,j}$ is positive, and the orientation of $M_{h,j}$ is induced from $W_{h,j}$ by the outward-normal-first convention of Relative fundamental class and boundary orientation.

Why the bundles exist. The clutching map $g_{h,j}$ is the restriction to $S^3$ of a polynomial map into ${\mathrm{Mat}}_{4\times 4}(\mathbb{R})$: for negative exponents use powers of $\bar{a}$. Its values on $S^3$ lie in $SO(4)$, so the clutching map is smooth and orientation preserving; over each closed hemisphere the bundle is trivial, and over the equatorial collar the two trivializations are compared by $g_{h,j}$. The smooth cocycle constructed this way satisfies the transition identities, so Construction of a vector bundle from a smooth cocycle produces a smooth rank-four vector bundle ${\xi }_{h,j}$ on $S^4$, whose total space has dimension $4+4=8$ (The total space of a rank-r bundle has dimension dim M + r). The quaternionic norm is preserved by $g_{h,j}$ because $N(a^{h}v{a}^j)=N(v)$ for $N(a)=1$, so it is constant along the clutching orbits and descends to a smooth bundle metric on ${\xi }_{h,j}$ (Every smooth vector bundle admits a smooth bundle metric); the disk and sphere bundles are taken with respect to this metric as in Disk bundle, sphere bundle, and Thom space: the differential topology interface.

Dimension and boundary. Fibrewise, $D({\xi }_{h,j})$ is the closed unit ball in ${\mathbb{R}}^4$ and its boundary is the unit sphere $S^3$; over the base $S^4$ this gives a smooth fiber bundle with fibre $D^4$ and boundary the corresponding $S^3$-bundle. Compactness follows from compactness of $S^4$ and of the closed unit ball, and the oriented smooth structures and boundary orientation are those fixed above. The sphere bundle $M_{h,j}$ is the total space of the fibration displayed, so the long exact homotopy sequence of a fibration applies to it. Nothing here asserts that $M_{h,j}$ is a homotopy sphere; that is proved separately under the Euler-number hypothesis $h+j=\pm 1$.