Standard and reflected two-sphere immersions have homotopic formal data in R^3

$E$ has a section: the differential $d\iota$ of the standard inclusion is a bundle monomorphism $T{S}^2\to {\epsilon }^3$ over $\iota$, hence a section $s_{\iota }$ of $E$ by [F2]: for the standard metrics $d{\iota }_x(v)=v$ is already isometric. Thus $\mathrm{\Gamma }(E)\mathrm{\ne }\mathrm{\varnothing }$. The fibre $V_2({\mathbb{R}}^3)$ is path connected by [F5] and ${\pi }_2(V_2({\mathbb{R}}^3))=0$ by [F4].

The section space $\mathrm{\Gamma }(E)$ is path connected: by the evaluation-fibration lemma [F3] applied with $m=2$, $n=3$, the long exact sequence makes ${\pi }_0\mathrm{\Gamma }(E)$ a quotient of ${\pi }_2(V_2({\mathbb{R}}^3))$ as soon as $\mathrm{\Gamma }(E)\mathrm{\ne }\mathrm{\varnothing }$, and that group is trivial by [F4]; equivalently the evaluation fibration is surjective on path components and its based section space has ${\pi }_0\cong {\pi }_2(V_2({\mathbb{R}}^3))=0$. Hence any two sections of $E$ are joined by a path of sections.

Consequently any two formal immersions $S^2\to {\mathbb{R}}^3$ lie in the same path component of $\mathrm{FImm}(S^2,{\mathbb{R}}^3)$: path components are the classes of the relation "joined by a continuous path" and a homotopy of formal data is a path in $\mathrm{FImm}$ [F6], so it suffices that by [F2] the space is homotopy equivalent to $C^{\mathrm{\infty }}(S^2,{\mathbb{R}}^3)\times \mathrm{\Gamma }(E)$ via fibrewise polar normalization and the first factor is contractible, so its path components are exactly those of $\mathrm{\Gamma }(E)$, a single point by step 2.1. Applying this to the two formal immersions of [F1] gives that $(\iota ,d\iota )$ and $(\iota \circ a,d(\iota \circ a))$ are homotopic through formal immersions.

For the difference-class description, first move both frame sections to one prescribed basepoint value by evaluation path lifting, using the path-connected fibre in [F5]. The evaluation lemma [F3] pulls them to the closed characteristic disk with the same, possibly nonconstant boundary map, and transports both disk models along a contraction supplied by a reference section. They then descend to based maps into $V_2({\mathbb{R}}^3)\cong \mathrm{S}\mathrm{O}(3)$; subtracting their classes in ${\pi }_2$ gives the difference obstruction. By [F4] this group vanishes, so the difference map is nullhomotopic and the two sections are homotopic, as already established in step 3.1. The two components of $O(3)$ are homeomorphic to $\mathrm{S}\mathrm{O}(3)$, so their second homotopy groups vanish too. Replacing $\iota$ by $r\circ \iota$ for a reflection changes the target by a rotation relative to $\iota \circ a$: for $r(x,y,z)=(x,y,-z)$ take $A=\mathrm{diag}(-1,-1,1)\in \mathrm{S}\mathrm{O}(3)$, and for any reflection $A=-r$ is likewise a rotation. A path of target rotations from $I$ to $A$ gives a homotopy of the corresponding formal data. Hence the reflected embedding has the same formal component as $\iota \circ a$.