Whitney–Graustein classification of plane circle immersions

Necessity: let $H$ be a regular homotopy from $f$ to $g$. By [F2] the formal data $(H_t,d{H}_t)$ form a continuous path in $\mathrm{FImm}(S^1,{\mathbb{R}}^2)$, so the winding invariant is constant along it by [F5]; for the derivative of an immersion the winding invariant equals the rotation number by [F3]. Hence $\mathrm{rot}(f)=\mathrm{rot}(g)$, and two curves with different rotation numbers are not regularly homotopic.

Sufficiency: assume $\mathrm{rot}(f)=\mathrm{rot}(g)$. By [F3] the winding invariants of the derivatives $(f,df)$ and $(g,dg)$ are equal, so these formal immersions lie in the same path component of $\mathrm{FImm}(S^1,{\mathbb{R}}^2)$, i.e. are homotopic through formal immersions. The derivative map is a weak homotopy equivalence by [F4], hence a bijection on path components, so $f$ and $g$ lie in the same path component of $\mathrm{Imm}(S^1,{\mathbb{R}}^2)$; by [F4] path components of $\mathrm{Imm}$ for the compact source $S^1$ are exactly the regular homotopy classes, so $f$ and $g$ are regularly homotopic.

Consequently rotation number gives a bijection ${\pi }_0\mathrm{Imm}(S^1,{\mathbb{R}}^2)\cong \mathbb{Z}$: necessity and sufficiency give well-definedness and injectivity. For $k\mathrm{\ne }0$, $f_k(\theta )=(\cos k\theta ,\sin k\theta )$ has velocity of norm $\mathrm{\mid }k\mathrm{\mid }$ and unit tangent $\mathrm{sgn}(k)(-\sin k\theta ,\cos k\theta )$, a constant rotation of $e^{ik\theta }$, hence degree $k$. For $k=0$, take $\delta (\theta )=(\cos \theta ,\sin 2\theta )$. Its velocity is $p(\sin \theta )$, where $p(u)=(-u,2-4u^2)$ never vanishes for real $u$: if its first coordinate is zero its second is $2$. The homotopy $p((1-t)\sin \theta )$ contracts this velocity loop to $(0,2)$ through nonzero vectors, so its degree is zero. Thus every integer occurs.

Reversing the domain gives ${\tau }_{f\circ a}(\theta )=-{\tau }_f(-\theta )$ for $a(\theta )=-\theta$. The constant target rotation by $\pi$ preserves degree and domain reversal negates it, so the reverse of a curve of rotation number $k$ has rotation number $-k$. In particular the once-traversed round circle and its reverse have values $1$ and $-1$ and are not regularly homotopic. The zero class is nonempty by step 3.1; its representatives are regularly homotopic to their reversals by step 2.1. Countable choice is inherited from [F4].