Can a sequence of degree one maps on, say, the unit circle, converges to a constant map in $W^{1,2}$ norm?


If the answer is yes, would you please provide an explicit example?



Can a sequence of degree one maps converges to a constant map in $W^{1,2}$ norm?
Lift your map to the universal cover to obtain a function $f:R\to R$ satisfying
$$f(x+1)=f(x)+1,$$
if the degree is $1$. 
If the Sobolev norm of $f-c$ is $<\epsilon$ then by Schwarz inequality
$$(f(1)-f(0))^2\leq\int\_0^1 (f'(t))^2dt<\epsilon,$$
which contradicts the first inequality if $\epsilon<1$.


