Euler number $\pm 1$ makes the Milnor sphere bundle a homotopy seven-sphere

Given: Integers $h,j$ with $h+j=\pm 1$ and the closed smooth seven-manifold $M_{h,j}$ of The Milnor sphere and disk bundles $M_{h,j}$ and $W_{h,j}$.

The Axiom of Choice is assumed (The Axiom of Choice); in particular ${\mathrm{A}\mathrm{C}}_{\omega }$ follows (AC implies DC implies countable choice).

If $h+j=\pm 1$, then $H_k(M_{h,j};\mathbb{Z})\cong H_k(S^7;\mathbb{Z})$ for every $k$ (Euler number $\pm 1$ implies the Milnor sphere bundle is a homology seven-sphere), and $M_{h,j}$ is closed, connected and simply connected (Milnor sphere bundles are simply connected, The Milnor sphere and disk bundles $M_{h,j}$ and $W_{h,j}$).

Under ${\mathrm{A}\mathrm{C}}_{\omega }$ a compact smooth manifold has the homotopy type of a finite CW complex (Compact smooth manifolds have finite CW models under countable choice).

Assume AC. For $r\ge 2$, an $(r-1)$-connected space with a supplied homotopy equivalence to a CW complex satisfies ${\pi }_r(X)\cong H_r(X;\mathbb{Z})$ by the absolute Hurewicz homomorphism; any class in $H_r$ therefore has a preimage represented by a based map $S^r\to X$ (Absolute Hurewicz theorem at the first nonzero degree).

Under AC, a homology equivalence $f:X\to Y$ between simply connected finite CW complexes is a homotopy equivalence. Indeed, cellular approximation and the finite mapping cylinder give a simply connected CW pair $(M_f,X)$ with zero relative homology by the homology pair sequence. Starting with its $1$-connectivity, relative Hurewicz inductively gives ${\pi }_r(M_f,X)=H_r(M_f,X)=0$ for every $r\ge 2$. The relative homotopy sequence makes $f$ a weak equivalence, and finite Whitehead makes it a homotopy equivalence (Cellular approximation for maps of CW pairs, Cellular mapping cylinders and relative cylinders are CW complexes, Long exact sequence of a pair, Relative Hurewicz theorem in the simple-connectivity range, Long exact sequence of relative homotopy groups, Whitehead theorem).