"Why can’t you have a retract of the disk onto its boundary circle?"

This is the question we will attempt to answer in this post.  Along the way, we will climb treacherous mountains and dive deep into dark waters — most frightening, perhaps: we will find things that we didn’t even know we were looking for.

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[This homology adventure will, unfortunately, have no pictures.  I want to make some, but my bamboo pad is being irritable.]

Here’s the game: Take the plane.  I remove, say, n points from the plane.  You tell me the homology groups of the resulting space.

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In this post, we’re going to do something that I thought helped me through homology, but that others no doubt think is a huge waste of time: we’re going to explicitly compute the homology groups of trivial or nearly-trivial things explicitly, without appealing to any theorems regarding homeomorphisms or the like.

Recall the things that we’ve done so far.  The steps to find the homology groups in this post (and, more generally, in life) will be as follows:

  1. Find the chain groups and their generators.
  2. Compute explicitly the image and kernel of the boundary maps.
  3. Find a nice way to express the kernel and image of these boundary maps by potentially using different, but equivalent, generators (we’ll get to this step soon; it’ll make more sense then).
  4. Find the quotient \displaystyle \frac{Ker(\delta_{i-1})}{Im(\delta_{i})}, which is equal to the (i-1)-th cellular homology group.
  5. Reflect on this solution: does it make sense?  Is it nice?  Do we love it?

So without further delay, let’s dig right in.

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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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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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Triangulating a Surface.

November 17, 2010

In some cases, we’d like to be able to break down a surface nicely into triangles or things which look like triangles.  There’s a (strong!) theorem which states that every compact surface has a finite triangulation and every surface has a (potentially infinite) triangulation.  We’ll talk about how to triangulate a surface below.

(Note: this used to be a part of the Homology Primer, but I decided against using triangulations to talk about homology.  Nonetheless, triangulation is a topic which comes up in topology so I decided to keep this post up as a reference.)

 

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Introduction to Simplices.

November 11, 2010

There are a number of ways in topology to make shapes.  We can use the euclidean plane and make them out of equations.  For example, we can make this torus:

image

by plotting the equation \displaystyle c - (\sqrt{x^{2} + y^{2}}) + z^{z} = a^{2} in x, y and z.

Another way we could think of the torus is taking one circle centered at some non-origin point on the xy-axis and rotating it about the x-axis, say.  You could think of this as taking one of those bubble wands, holding it at arms length, and spinning around in a circle.  If the bubble didn’t pop, you’d make a torus bubble around you.  Wouldn’t that be cool?

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Let me note two things here — one is a mathematical point, one is a technical point. 

First, math: the type of homology I will be introducing here will be cell homology, because I think that it’s the best way for someone to actually get their hands dirty and compute homology groups of spaces.  This is not too much of a loss of generality, since in nice spaces (eg, finite CW complexes) this is the same as most of the other homology theories. 

Now, a technical note.  I am now a beginner user of the bamboo pen tablet, which, so far, is fantastic.  This means that many of my new pictures (whenever possible) will be hand-drawn.  Note that when precision counts, I will continue to use mathematica, but generally drawing things in mathematica is a huge pain past just graphing equations. 

Because my drawing is terrible, in general, if you have any questions about what the pictures mean, please comment and I will try to elaborate.  What seems obvious to me is not necessarily obvious to all of you, so telling me that my drawing of a hexagon looks like a crying cat will help me teach better.

Now onwards to cells!