In about 23 hours, I’ll step on to the train in Jena, heading for T’bilisi, Georgia.
On Monday, I’ll give a talk on my research into -structures in group cohomology. If you’re curious, I already put the slides up on the web.
I’ll try to blog from T’bilisi, but I don’t know what connectivity I’ll have at all.
Is up at the GeomBlog.
This fortnight has a lot of goodies, among those a call for reading Grothendieck and a blogpost by Ian Stewart.
I have been somewhat remiss in announcing these lately – but over at nOnoscience, the 7th Carnival of Mathematics just got posted.
I’m featured again – as are many other very readable bloggers. Go. Read.
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 .