Suppose (Ln) is a sequence of loops in a torus S1 × S1 converging in the Hausdorff metric to some set L in the torus. Suppose also that for each loop Ln the projection map p:S1 × S1 -> S1 defined by p(x,y) =x when restricted to Ln is not null-homotopic, then can we conclude that the restriction of the projection map p to L is also not null homotopic? Or are there counter examples?



limiting behaviour of converging loops on a torus
Your intuition is correct: the map p : L → *S*1 is not nullhomotopic.  



If p : L → *S*1 was nullhomotopic, then it would factor through the universal cover ℝ of *S*1.  

The inclusion ι : L → *S*1 × *S*1 would therefore factor through ℝ × *S*1.  

Let ι' : L → ℝ × *S*1 denote a lift of ι,
and let L(0) := ι'(L).  

The subset L(0) ⊂ ℝ × *S*1 is then one of the connected components of the preimage of L in ℝ × *S*1.  

Let us call the other components L(*n*) for *n* ∈ ℤ.  



Let *C*0 ⊂ ℝ × *S*1 be a loop in that
separates L(0) from L(1).  

We may also assume that *C*0 is not nullhomotopic in ℝ × *S*1 [this still needs a small argument...].


The projection *C* ⊂ *S*1 × *S*1 of *C*0 is then a loop in the complement of L,  
 and
represents the element (0,1) of ℤ×ℤ = π1(*S*1 × *S*1).


Since L lies in the complement of *C* and L*n* → L in the Hausdorff metric,  
 there exists an *n* such that L*n* lies in the complement of *C*.  

This contradicts your assumtion that p : L*n* → *S*1 is nullhomotopic.


