There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:


•  **Continuous maps:**
Using the [unique path lifting property](http://en.wikipedia.org/wiki/Homotopy_lifting_property) of the universal covering map $\mathbb R\to S^1$, any continuous map $\gamma:S^1\to S^1$
can be lifted to a path $\tilde\gamma:[0,1]\to \mathbb R$. The winding number of $\gamma$ is then defined by the formula $$W(\gamma)=\tilde \gamma(1)-\tilde \gamma(0).$$


• **Sobolev-1/2 maps:**
If we view a map $\gamma:S^1 \to S^1$ as a special case of a function $S^1\to \mathbb C$, then the winding number is the algebraic area (I'm omitting all factors of $\pi$ here) enclosed by $\gamma$. By Stokes' theorem, this is the integral over $S^1$ of the 1-form $\gamma^\*(-ydx+xdy)$. The latter is clearly quadratic in $\gamma$. If $\gamma=\sum\_{n\in \mathbb Z}\gamma\_nz^n$
is the Fourier series of $\gamma$, then each loop $z\mapsto \gamma\_nz^n$ encloses an area of $n|\gamma\_n|^2$ and the cross-terms don't contribute anything. Therefore, we get the formula
$$
W(\gamma)=\sum\_{n\in \mathbb Z}n|\gamma\_n|^2.
$$
At this point, recall that $\sum\_{n\in \mathbb Z}|n+1||\gamma\_n|^2$ is the definition of (the square of) the Sobolev-1/2 norm of $\gamma$. Therefore, $\gamma$ having finite Sobolev-1/2 norm is the obvious condition for the above sum to converge.


---


Now, it is well known that
$$
\{\text{Continuous}\} \quad\not\subseteq\quad \{\text{Sobolev-}1/2\}
$$
and 
$$
\{\text{Sobolev-}1/2\} \quad\not\subseteq\quad \{\text{Continuous}\}.
$$
Thus my question:



> 
> Is there a reasonable class of maps $S^1\to S^1$ that contains both the continuous maps and Sobolev-1/2 maps, and on which the winding number makes sense?
> 
> 
> 



For which maps $S^1\to S^1$ is the winding number defined?
A class of maps including both continuous and $H^{1/2}$, where an extension is available, is *Vanishing Mean Oscillation*, [VMO](http://en.wikipedia.org/wiki/Vanishing_mean_oscillation#The_space_VMO). This has been treated by several authors starting I think with Haïm Brezis. You can find quite a lot googling "degree theory for VMO maps".


