Question

Suppose (L_n) 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 L_n the projection map p:S¹ × S¹ -> S¹ defined by p(x,y) =x when restricted to L_n 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?

Answer 1

Your intuition is correct: the map p : L → S¹ is not nullhomotopic.

If p : L → S¹ was nullhomotopic, then it would factor through the universal cover ℝ of S¹.
The inclusion ι : L → S¹ × S¹ would therefore factor through ℝ × S¹.
Let ι' : L → ℝ × S¹ denote a lift of ι, and let L₍₀₎ := ι'(L).
The subset L₍₀₎ ⊂ ℝ × S¹ is then one of the connected components of the preimage of L in ℝ × S¹.
Let us call the other components L_(n) for n ∈ ℤ.

Let C₀ ⊂ ℝ × S¹ be a loop in that separates L₍₀₎ from L₍₁₎.
We may also assume that C₀ is not nullhomotopic in ℝ × S¹ [this still needs a small argument...].

The projection C ⊂ S¹ × S¹ of C₀ is then a loop in the complement of L,
and represents the element (0,1) of ℤ×ℤ = π₁(S¹ × S¹).

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 → S¹ is nullhomotopic.

Answer 2

EDITED. The following example should not be conisdered as a contre-example, it is just a well known example of a pathology that can happen.

EXAMPLE. $L$ is composed of a union of the vertical circle $(0, S^1)$ and a disjoint $R^1$ that is emdedded in $T^2$ in such a way, that it accumulates to $(0, S^1)$ from both sides. This $R^1$ projects one-to one to the horisontal circle without a point and can be represented a as graph (function of $x$), than if is given by the following formula:

$y= (sin(1/x))$

The point is that such $L$ can be Hausodrff approximated by a sequence of circles $s_n$ that are not null-homotopic ($s_n$ wiggles more an more near the vertical circle $(0, S^1)$ as $n\to \infty$).

The problem with this example, is that the topology on the union of $S^1$ and $R^1$ that we should take (I guess) is the toplogy induced from $T^2$. And for this induced topology, I guess the projection to the horisontal cicle is non null-homotopic...