Recognise the shape

1 · State the result

Beginner

What does Cameron Gordon prove?

That a powerful enough computer will eventually recognise every four-dimensional shape That no algorithm can recognise the four-dimensional sphere, S⁴ That no algorithm can tell whether a given shape is the connected sum of 7 copies of S² × S²

2 · Place the sphere

Intermediate

In which dimensions can the sphere be recognised by an algorithm?

In dimensions 2 and 3 In dimension 5 and above In dimension 4 only

3 · Find the open case

The unknotting problem asks whether an n-dimensional sphere, embedded in the (n + 2)-dimensional sphere, is unknotted.

Expert

For which value of n is this problem still open?

n = 1, ordinary knots n = 3 and above n = 2, spheres in S⁴

4 · Follow the reasoning

For every word w in a group whose word problem is unsolvable, Gordon builds a shape X(P_w). If w equals the identity, X(P_w) is #7(S² × S²); otherwise, its group of loops is non-trivial.

Expert

Why does this make recognising #7(S² × S²) impossible?

Because the shapes X(P_w) are too large for a computer Because a recognition algorithm would solve this group's word problem Because all the shapes X(P_w) have the same group of loops

Shape recognised… or not

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