Innocent Problem?

I thought of a random topological problem a week ago in my Analysis class and it has been bugging me for quite a while. I tried searching around but couldn’t find anything.

Consider a non-intersecting curve \gamma: I=[0,1] \to \mathbb{R}^{3}. It can even be a closed loop but it shouldn’t be ‘too straight’ else the problem is trivial. Considering a ‘thickening of the curve’. If X(0) is the curve at time t=0 and X(t) be obtained by adjoining closed disks of radius t at every point of the curve \gamma(I won’t even bother to formalize it). Now, consider the relative complement homology(aka local at subspace) \tilde{H}_{k}(\mathbb{R}^{3},\mathbb{R}^{3} \backslash X(t)). The vague problem is to find nice examples of curves and how their homology groups vary with the parameter t. For example, if one considers an 8-loop except it doesn’t exactly intersect(instead, there is a small gap), then at t=0, we obviously have the homology of a circle. If we thicken it a little bit, we’ll get a 4-torus which gains information in H_{2}(\mathbb{Z}^{4}) and thicken it even more, the entire thing collapses to 0. The same thing works with a circle(you get a torus on thickening). For a trivial example, if you just take a line segment then any thickening deformation retracts to the segment which contracts to a point.


I asked my Analysis professor about it, unfortunately, he didn’t have much of an idea and directed me to faculty who may know something about it(and who I’ve yet to contact). Then, I approached my linear algebra professor after class to ask if he knew anything about it(probably a bad idea since he works in representation theory), he just laughed and walked out. I have some stuff jotted down on the problem making little(obvious) progress but I wanted to gather more examples of classes of curves and some inkling of a theorem before I post anything.

I am currently working on the third part of the Derived Functors sequence, some interesting things in stable homotopy theory.

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