Lower-dimensional sphere maps are based nullhomotopic

Triangulate $S^d$ by the boundary of the cross-polytope: faces are convex hulls of signed coordinate vectors containing no opposite pair. Its realization is $\sum \mathrm{\mid }{x}_i\mathrm{\mid }=1$. Radial normalization $x\mathrm{/}\mathrm{\parallel }x{\mathrm{\parallel }}_2$ and its inverse $y\mathrm{/}\sum \mathrm{\mid }{y}_i\mathrm{\mid }$ identify it with the sphere. To put any specified basepoint b at a vertex, if b≠e1 use $H_b(x)=x-2⟨x,e_1-b⟩(e_1-b)\mathrm{/}\mathrm{\parallel }{e}_1-b{\mathrm{\parallel }}^2$; the identity is used if b=e1. Its orthogonality follows by expanding the inner product, and $H_b(e_1)=b$ since $\mathrm{\parallel }{e}_1-b{\mathrm{\parallel }}^2=2(1-b_1)$. Thus both spheres have finite triangulations with their basepoints vertices, including the two-point S0.

Apply F1 to the pair consisting of the source sphere and its singleton vertex, mapping to the target sphere and its singleton vertex. The pair homotopy fixes a, because its image there must stay in {b}. The simplicial image has dimension at most k. Since k<r it misses the interior of every top-dimensional target simplex; choose p to be the radial image of the barycenter of one such simplex. Then p is omitted and p≠b.

On $S^r\setminus \{p\}$, set $q(x)=(x-⟨x,p⟩p)\mathrm{/}(1-⟨x,p⟩)\in p^{\perp }$. Its inverse is $v\mapsto (2v+(\mathrm{\parallel }v{\mathrm{\parallel }}^2-1)p)\mathrm{/}(\mathrm{\parallel }v{\mathrm{\parallel }}^2+1)$: substitution uses $⟨v,p⟩=0$ and gives both identity composites. The denominators are positive on the specified domains. By F2 the affine homotopy $(v,t)\mapsto (1-t)v+tq(b)$ is continuous and fixes q(b). Composing with the simplicial image and q inverse gives a based nullhomotopy. Prepend the based approximation homotopy from step 2.1, pasting on the two closed time halves. This proves the claim, including k=0.