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 the loop can be contracted to a point and hence form a sphere (see Poincaré conjecture).

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.

In the next tutorial, you will learn about determinism vs non-determinism.