Two questions usually immediately spring to mind when you are introduced to some mathematical topic: "why should I care?" and, assuming you do care, "how do I use it?"  With homology, we can either have a serious answer or a silly answer; I prefer the latter, so whenever people ask me what I do, I tell them I spent five years studying math so that I can officially say that a donut has one more hole than a sphere.

If you already know about the fundamental group, then you might be saying to yourself, "Alright, I already know how to tell things with holes apart.  We have the fundamental group for that.  And if the fundamental group doesn’t work, then we have higher homotopy groups.  Why do we need homology?"  It turns out that homotopy groups are actually quite difficult to calculate, even for the most simple structures: n-spheres.  In fact, it turns out that even if our sphere is only n dimensional, we can have nontrivial \pi_{r}(S^{n}) for r > n.  That’s kind of crazy!  We’d like it if our topological invariant was a little bit easier to handle.

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When I began doing actual calculations for the homology groups, I was surprised by how simple the idea was given the right structures: in the finite case it reduces to finding the kernel and image of a few matrices.  This is in direct contrast to how I originally learned the subject: axiomatically.  What I’ve described below is how to begin building CW-complexes.  These structures are a nice way to begin learning how to find homology groups (in this case, we call them “cellular homology groups”, but they turn out to be equivalent to most of the other homologies you’ve probably heard of in the case of CW-complexes) and are relatively simple and intuitive to describe.  Let’s begin by talking about the basic building blocks of CW-complexes: cells.

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Reader Beware.

I planned to do a post about tensor products (what they are, why we should care, what we do with them, etc.) but because I’m not comfortable with all of that quite yet, I’m going to assume you know what tensor products are, and do a few explicit calculations.  So, in short, if you don’t already know what tensor products are, don’t read this post.

Our notation will be as follows: k is a field, R is a commutative ring with 1\neq 0, and \otimes_{R} will denote the tensor product of modules over a ring R.  As usual, R[x] will denote the polynomials in x with coefficients in R.

(Note:  My thanks to Brooke, who pointed out that I kept writing "+" when I meant "\otimes."  I hope I’ve not made this error elsewhere, as tensors are "pretty different" from standard addition.)

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(I’ve decided against giving a proof of Rouché’s theorem until such a time as I find one that doesn’t use algebraic topology or isn’t tedious as hell.)

 

Let’s simply state Rouché’s theorem, and then we’ll talk about how to actually apply Rouché’s theorem.

 

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(In this part: The Argument Principle and the Winding Number.)

Each of these three theorems (the argument principle, the winding number theorem, and Rouché’s theorem) are all interesting in their own right, but something really special happens when you put them into a cocktail mixer and shake them up together.  Really; I’m not a fan of analysis, but what we’re doing in this post I think of as almost magical

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EDIT: It seems that scribd is now behind a paywall now.  :(  

Brown and Churchill (8th ed) was the book I used for the second complex analysis class I’ve had to take so far (the first was Lang).  My class went over the first six chapters and half of the seventh: so, up to the middle of the section on applications of residues.

To prep for the final, I compiled a quick, slightly-shorter list of things that I feel the complex student should know if they’ve used this book and have gotten to around the same point.  I’ve excluded the chapter on applications of residues, since it’s a relatively short chapter with better pictures in the text than ones I could draw at 5am.  Because sharing is caring, below is a link to the pdf.  Enjoy!

http://www.scribd.com/doc/45170305

Sometimes, we can read a whole bunch of math and not get it until a simple picture is drawn.  For example, the notion of upper semi-continuous was not clear to me (at all) until a picture was drawn.  The "idea" of the mean value theorem is similar — drawing the tangent and parallel secant really shows what the theorem is trying to say. 

This post is to say in words, and then show in pictures, why we should have the fundamental group about a basepoint x_{0} of {\mathbb R}^{n} be trivial.  In other words, we’d like to prove \pi_{1}({\mathbb R}^{n}, x_{0}) = 0

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If you go to a math-savvy high school student and ask them what the complex plane looks like, they’ll probably say something like, “It’s like the xy-plane with the real values on the x-axis and the imaginary values on the y-axis.”  If you go to someone studying complex analysis and ask them what the complex plane looks like, they might say something a little more surprising: “Think of it as a sphere where the top point is infinity.”

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There was a question on my topology final that asked: in a Euclidean space prove that an open subset U is connected if and only if it is path connected.  Since this is not a difficult problem and it is available in a number of textbooks, I’m going to answer it with lots and lots of details and draw a few pictures, since I like drawing pictures.

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