[FOM] Re: Shapiro on natural and formal languages
Timothy Y. Chow
tchow at alum.mit.edu
Wed Dec 1 09:47:08 EST 2004
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
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