Affine Clifford–Hopf maps are highly connected

The terminal winding is $w=p\mathrm{/}4-\eta \ge 0$, and $s_l=s_k$, so $p-4\mathrm{\mid }w\mathrm{\mid }=4\eta \ge 4d{s}_l$. Apply [F2] to shorten the variable second path, fixing the prescribed parameter boundary. A minimal second path is determined by an invariant splitting ${\mathbb{R}}^p=L_0\oplus L_1$, with midpoint $+J_k$ on $L_0$ and $-J_k$ on $L_1$. Its determinant winding gives ${\dim }_{\mathbb{C}}{L}_1=\eta$. Combining with [F1] yields a $d$-connected map from the splitting space into the sphere mapping space.

On $L_0$ the terminal loop is a path followed by its reverse. Contract it by replacing $\gamma (t)$ by $\gamma (st)$, $1\ge s\ge 0$, before taking that concatenation. This is a based contraction continuous in the splitting projection. On $L_1$ the loop is the full positive Clifford circle. After the earlier suspensions the latter becomes the positive linear Hopf map, by expanding the cosine and sine suspension coordinates.

There is an additional correction on $L_0$: iterated suspension of its constant loop is a folded hemisphere map, not literally a constant sphere map. Its image lies in the closed hemisphere of the great sphere spanned by $I,J_1,\dots ,J_l$, with nonnegative $J_l$ coefficient, a product of nonnegative suspension sine factors. Write a point as $(y,a)$ with $a\ge 0$ and $\mathrm{\mid }y{\mathrm{\mid }}^2+a^2=1$. Stereographic projection from $-J_l$ gives $z=y\mathrm{/}(1+a)$ in the closed unit ball, with $I$ at $e_0$. Set $z_s=(1-s)z+s{e}_0$ and $y_s=\frac{2z_s}{1+\mathrm{\mid }{z}_s{\mathrm{\mid }}^2},a_s=\frac{1-\mathrm{\mid }{z}_s{\mathrm{\mid }}^2}{1+\mathrm{\mid }{z}_s{\mathrm{\mid }}^2}\mathrm{.}$ This contracts the hemisphere to $I$ while fixing $I$. Every sphere point remains an orthogonal Clifford matrix. Apply this on $L_0$ and keep the Hopf map on $L_1$. The universal scalar formulas and invariant projections make the homotopy continuous in the splitting.

The invariant splitting space is a compact Grassmannian. Its projection $P_1$ is recovered from the affine map by $P_1=(I-\varphi (-e_0))\mathrm{/}2$. Thus splitting-to-affine-map is a homeomorphism onto the affine subspace. Step 2.1 and step 3.1 show that the $d$-connected map of step 1.1 is homotopic to that homeomorphism followed by the actual affine inclusion. Homotopy invariance of relative homotopy groups therefore makes the inclusion $d$-connected. Equivalently, extend the resulting relative disk homotopies over cells, using [F3]. This last argument keeps arbitrary prescribed affine boundary maps fixed; it does not infer connectivity merely from a set-theoretic image.