A starting place

Homotopy in wild spaces

If I’m thinking about wild topology, I’m probably thinking about topological spaces (geometric-type objects) which have points where non-trivial structure is visible no matter how far you zoom in. Though I’m not so interested in generating fractals, these pretty little things do reach the point of being “complex on an infinitely small scale.”

Since we are talking about homotopy, we are allowed to bend and stretch the objects (but not cut or puncture them) and consider it to be the same object. The object of homotopy theory is to classify spaces up to homotopy equivalence: Two spaces X and Y are homotopy equivalent if there are maps f\colon X \to Y and g\colon Y\to X such that the compositions gf\simeq id_{X} and fg\simeq id_{Y}.

When spaces look “simple” on a local level (like manifolds or CW-complexes), this classification boils down to using discrete gadgets (like homotopy groups) that look for algebraic structure hiding within. Things become much more complicated when you begin to allow local pathologies; if your space has important structure as you zoom in you’d better make sure it’s still there when you zoom in on your newly deformed object!

The tip of the iceberg

Even in the usual Euclidean plane we learn about in middle and high school it is all too easy to construct such objects. One of the simplest examples is the Hawaiian earring which can be constructed by taking a sequence of circles that get smaller and smaller and all converge up on a single point of intersection.

The Hawaiian earring \mathbb{HE}

We can realize the Hawaiian earring as a compact planar set: If

C_n=\left\{(x,y)\in \mathbb{R}^2|x^2+\left(y-\frac{1}{n}\right)=\frac{1}{n^2}\right\},

then

\mathbb{HE}=\bigcup_{n\geq 1}C_n

so the point (0,0) is the intersection \bigcap_{n}C_n.

Note the Hawaiian earring is distinct from the countable wedge of circles \bigvee_{n\geq 1}S^1 only at a single point. In fact, there is a continuous bijection

\bigvee_{n\geq 1}S^{1} \to \mathbb{HE}

which is a local homeomorphism at every point except the intersection point. This one point makes a world of difference. For one, the Hawaiian earring is compact and \bigvee_{n\geq 1}S^1 is not. For another, the two are not homotopy equivalence; we’ll investigate the details later.

Despite the simple appearance, \mathbb{HE} has an enormous amount of combinatorial group theory and topological algebra hiding within…and this is pretty much the simplest example of a space where covering space theory falls apart.

The rest

More generally, you could ask: What does it take to classify connected subspaces of the plane up to homotopy equivalence? What if we add conditions like locally path connectedness or compactness? What if we extend to subspaces of \mathbb{R}^{3}?

One of the blossoming approaches to these questions makes use of a classical tool in topology: the fundamental group \pi_1. This is likely to be the topic of upcoming posts if I can manage to find the time to write them…

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