[FOM] Re: Shapiro on natural and formal languages

[Timothy Y. Chow]

On Tue, 30 Nov 2004 JoeShipman at aol.com wrote:
> Chow proposes two examples:
[...]
> Unfortunately, example 1 is too easy to prove non-visually.
> Example 2 doesn't really qualify

All right, let me try again:

- The fundamental group of the punctured torus is the free group on two
   generators.

Proof: Stick the fingers of both hands into the puncture and pull back the 
outer skin all the way around the torus, as if you're opening a curtain.
This shows that the punctured torus is homotopic to a wedge of two circles.

There are lots of similar "rubber sheet" arguments with a similar flavor. 
Here's a fun one: Can a two-holed torus where the two loops are "linked"
be unlinked in R^3?


- pi_3(S^2) is nontrivial.

Proof: This amounts to visualizing the Hopf fibration.  There's a picture 
of this in the Thurston-Levy book on 3-dimensional topology.  Francis Su 
has also described a slightly different picture to me.

This example is interesting because S^3 can't be embedded in R^3, so not 
everyone finds it easy to visualize what's going on.  So maybe the formal 
proof is easier or more convincing for many people.  However, I believe 
that if you are able to form the visual picture, then you gain insight 
that you might not get from the formal proof.  It would be fun to ask 
various topologists for their opinions on the visual versus the formal 
presentation of the Hopf fibration.

Tim