Tutorial 8: Holes, Tunnels and Spheres

In the last tutorial, you learned that Hypertorus Homotopy is a way to interpret Homotopy Theory using normal paths:

f[g]

Here, g is a loop around a point f.

Now, let us revisit the imaginary normal path in Tutorial 6:

concat[len x len -> id]

Using the Hypertorus Homotopy interpretation, len x len -> id is a loop around a point concat.

However, since this normal path has no solution, the loop around the point is not contractible.

This means, that the loop defines a hole.

Just like points, two holes can have a homotopy, for example:

concat[(id . len) x (id . len) -> id] <=> concat[len x len -> id]

In physics of space-time, this corresponds to a wormhole.

Reasoning about holes can also be used for toruses:

f[g[h]]

Here, f is a point, g is a loop and h is a torus.

When g[h] has no solution, the loop g hollows out the torus h.

A strange thing is that f[g[h]] can have a solution, even though g[h] has no solution. This is because f can be an imaginary normal path.

When f[g[h]] has a solution while g[h] does not have a solution, the hollow torus might be thought of as a tunnel loop.

Two hollow toruses can have a homotopy:

f1[g1[h1]] <=> f2[g2[h3]]

A more complex example: Spheres are holes with loops that can be contracted to a point:

f[g1][g2]

Here, f[g1] has no solution, which creates a hole. The loop g2 closes the hole such that it can be contracted to a point and hence form a sphere.

In the example used before:

∵ concat[len x len -> id][id x id -> len]
∴ concat[(id . len) x (id . len) -> (len . id)]
∴ concat[len x len -> len]
∴ concat[len]
∴ add

Here, concat[len x len -> id][id x id -> len] is a sphere.

At first, it might seem counter-intuitive that a sphere can behave like a point. However, from a topological perspective, when one can only contract paths in some space to check for holes, a sphere might be indistinguishable from a point.

A sphere can be continuously transformed into a point. The transformation of the sphere into a point is a homotopy.

In summary: One can reason about holes and other similar objects such as tunnels and spheres using normal paths in the same way one can reason about points.