Michi’s blog » archive for '9th grade topology'

Young Topology: The fundamental groupoid

  • May 4th, 2007

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

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 Jänich, 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)

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Michi is a recent PhD working in homological algebra and applied algebraic topology. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
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