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

## 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 , if there is a continuous map 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 , such that and .

Two spaces X,Y are homotopy equivalent if there are maps , such that and .

## 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