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Young Topology: The fundamental groupoid

  • May 4th, 2007
  • 3:26 pm

Today, with my bright topology 9th-graders, we discussed homotopy equivalence of spaces and the fundamental groupoid. In order to get the arguments sorted out, and also in order to give my esteemed readership a chance to see what I’m doing with them, I’ll write out some of the arguments here.

I will straight off assume that continuity is something everyone’s comfortable with, and build on top of that.

Homotopies and homotopy equivalences

We say that two continuous maps, f,g:X→Y between topological spaces are homotopical, and write f\simeq g, if there is a continuous map H\colon X\times[0,1]\to Y such that H(x,0)=f(x) and H(x,1)=g(x). This captures the intuitive idea of step by step nudging one map into the other in formal terms.

Two spaces X,Y are homeomorphic if there are maps f\colon X\to Y,f^{-1}\colon Y\to X such that ff^{-1}=\operator{Id}_Y and f^{-1}f=\operator{Id}_X.

Two spaces X,Y are homotopy equivalent if there are maps f\colon X\to Y,f^{-1}\colon Y\to X such that ff^{-1}\simeq\operator{Id}_Y and f^{-1}f\simeq\operator{Id}_X.

This is a preview of Young Topology: The fundamental groupoid. Read the full post (840 words, 46 images)

Bright students and topology

  • March 2nd, 2007
  • 3:39 pm

Today, I started an experiment together with the local specialised secondary school. I’ll be taking care of two of their brightest students, meeting them roughly once a week, and taking them on a charge through algebraic topology. At the far end shimmers knot theory and other funky applications; and on the way there, I hope for many interesting spinoffs and avenues.

They got, today, Armstrong’s Basic Topology, and an extract from the German topology book by J�nich, and on monday, we shall go over the formal definition of topological spaces, and of continuous functions together.

I plan to keep updates on our progress here on the blog – with the questions I send them off with each meeting as well as some sort of discussion about how this setup is working out, if at all.

For the first trip, the questions I dumped in their laps were:

  • What topologies are possible on the set {0,1}?
  • What topologies are possible on the set {0,1,2}?
  • Which are the continuous functions between the topologies above?
  • Give an example of a continuous and a discontinuous function each for the following cases
    1. f\colon\mathbb R\to\mathbb R with the standard topology on \mathbb R
    2. f\colon\mathbb C\to\mathbb C with the standard topology on \mathbb C
    3. f\colon\mathbb Z\to\mathbb Z with the discrete topology
    4. f\colon\mathbb Z\to\mathbb Z with the finite-complement topology
(206 words, 6 images)