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 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?
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 $\epsilon<1$.