I seem to have become the Goto-guy in this corner of the blogosphere for homological algebra.
Our beloved Dr. Mathochist just gave me the task of taking care of any readers prematurely interested in it while telling us all just a tad too little for satisfaction about Khovanov homology.
And I received a letter from the Haskellite crowd – more specifically from alpheccar, who keeps on reading me writing about homological algebra, but doesn’t know where to begin with it, or why.
I have already a few times written about homological algebra, algebraic topology and what it is I do, on various levels of difficulty, but I guess – especially with the carnival dry-out I’ve been having – that it never hurts writing more about it, and even trying to get it so that the non-converts understand what’s so great about it.
So here goes.
- February 13th, 2006
- 1:55 pm
- February 12th, 2006
- 11:14 pm
If I’m going to take this renewed interest in trying to understand differential geometry more seriously, I might as well read more than one source on it. So, I’ll start a sequence of posts on this book as well.
Just as Merkulov, Lee starts with a short definition of what a topological manifold is, including definitions for the terms needed for the treatment.
An n-dimensional topological manifold is a second countable Haussdorff space of local Euclidean dimension n.
Next, Lee goes on to define coordinate charts. I won’t repeat the treatment, since he doesn’t really bring anything Merkulov hasn’t talked about in some manner or other. Atlases, smooth structures and equivalence of atlases also merits some treatment.
- February 11th, 2006
- 11:03 pm
- February 11th, 2006
- 2:23 pm
I’ll do this in posts and not pages on further thought…
Sergei Merkulov at Stockholm University gives during the spring 2006 a course in differential geometry, geared towards the algebra graduate students at the department. The course was planned while I was still there, and so I follow it from afar, reading the lecture notes he produces.
At this page, which will be updated as I progress, I will establish my own set of notes, sketching at the definitions and examples Merkulov brings, and working out the steps he omits.
Familiar parts in unfamiliar language
Merkulov begins the paper by introducing in swift terms the familiar definitions from topology of topology, continuity, homeomorphisms, homotopy, and then goes on to discuss homotopy groups, and thereby introducing new names for things I already knew. Thus, I give you, for a pointed topological space