Refuted: the figure-eight and the round circle are regularly homotopic as oriented immersions

$\delta (\theta )=(\cos \theta ,\sin 2\theta )$ has velocity $(-\sin \theta ,2\cos 2\theta )$, whose squared norm ${\sin }^2\theta +4{\cos }^{2}2\theta$ never vanishes: if $\sin \theta =0$ then $\cos 2\theta =1$. It is therefore an immersion. If $\delta (\theta )=\delta (\varphi )$, equality of cosines gives $\varphi =\theta$ or $\varphi =-\theta$ modulo $2\pi$. In the latter case $\sin 2\theta =0$; the only distinct pair is $\pi \mathrm{/}2,3\pi \mathrm{/}2$, both mapping to $(0,0)$. Thus the origin is its unique double point. The round circle is an injective immersion with unit-speed velocity.

The velocity of $\delta$ is $p(\sin \theta )$ with $p(u)=(-u,2-4u^2)$. The map $p$ never vanishes, and $p((1-t)\sin \theta )$ contracts the velocity loop to $(0,2)$ through nonzero vectors. Hence $\mathrm{rot}(\delta )=0$. The unit tangent of ${\gamma }_1$ is $(-\sin \theta ,\cos \theta )$, a rotation of the identity circle map and of degree $1$.

Any smooth homotopy $H:S^1\times [0,1]\to {\mathbb{R}}^2$ through immersions has continuous nonzero velocity ${\mathrm{\partial }}_{\theta }H(\theta ,t)$, so its normalized velocity $\tau (\theta ,t)$ is a continuous homotopy of circle maps. In complex notation ${\beta }_t(u)=\tau (2\pi u,t)\mathrm{/}\tau (0,t)$, $0\le u\le 1$, is a based circle loop, and $(u,t)\mapsto {\beta }_t(u)$ is a homotopy fixing both endpoints at $1$. By [F1] its degree is $\mathrm{rot}(H_t)$, since the target rotation used to base it does not change degree. By [F3] these degrees agree at the two ends. As step 1.2 gives $0\mathrm{\ne }1$, no such regular homotopy joins $\delta$ to ${\gamma }_1$, refuting the false claim without a choice hypothesis or the sufficiency direction of classification.

The obstruction also distinguishes maps with the same image: ${\gamma }_2(\theta )=(\cos 2\theta ,\sin 2\theta )$ is an immersion with the round circle as its image and no transverse self-intersection, yet its unit tangent $(-\sin 2\theta ,\cos 2\theta )$ has degree $2$. The invariance argument of step 2.1 therefore separates it from ${\gamma }_1$ as well. Thus neither the image nor its number of transverse self-intersections determines the regular homotopy class; rotation number separates the figure-eight from the round circle.