Abstract Nonsense

Crushing one theorem at a time

The Fundamental Groupoid and Group (Pt. IV)


Point of Post: This is a continuation of this post.

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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 4 Comments

The Fundamental Groupoid and Group (Pt. III)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 1 Comment

The Fundamental Groupoid and Group (Pt. II)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 1 Comment

The Fundamental Groupoid and Group (Pt. I)


Point of Post: In this post we describe the fundamental group and groupoid functors.

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Motivation

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We have discussed in previous posts how the notion of homotopy as being an equivalence that captures, and focuses in on, the various notions of connectivity that a space carries. Now, the first thing one does when one defines a new notion of equivalence is to try to classify all objects of interest up to this equivalence. For us, this means that we would (in a perfect world) be able to classify all spaces up to homotopy equivalence. Of course, as in most categories(subjects of study), this is an untenable goal. It is not practically possible to, say, explicitly classify all finite groups (even though we have made stupendous strides in this direction).

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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 2 Comments

Homotopy and the Homotopy Category (Pt. III)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , | Leave a comment

Homotopy and the Homotopy Category (Pt. II)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , | 1 Comment

Homtopy and the Homotopy Category (Pt. I)


Point of Post: In this post we motivate, define, and discuss the notion of homotopy. We then introduce the homotopy category as a quotient category of \mathbf{Top}.

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Motivation

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In topology we care about the geometry of a space–how we can describe all of the geometric properties that a space has. For example, we care if the space is compact, we care if it locally connected, we care if it T_4. All of the properties factor into whether or not two given topological spaces are to be considered “the same”.  That said, for large portions of mathematics some of the topological invariants of a space are less important than others. This can occur either because they literally matter less to us (for example, compactness is something that, while nice, isn’t an absolute necessity for a topological space to be nice). This can also be true because the subject area we are are working in contains spaces which necessarily already satisfy some of the properties (for example, metric spaces already satisfy all of the nice separation axioms).

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August 29, 2012 Posted by | Algebraic Topology | , , , , , | 1 Comment

Singular Homology (Pt. II)


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February 21, 2012 Posted by | Algebraic Topology, Topology | , , | Leave a comment

Singular Homology (Pt. I)


Point of Post: In this post we discuss the notion of the singular homology of a topological space.

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Motivation

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Our immediate goal right now is to discuss homological algebra. That said, it would be an absolute crime to discuss such an abstruse concept without the slightest idea where the algebra came from! In particular, we offer in this post both motivation for why one would study homological algebra, as well as a good “model” for which we can test our homological results against. Now, I don’t want to act as though we are doing singular homology just for the sake of examples–it is the beginning to an extremely important branch of mathematics, that being algebraic topology of course. In fact, from pretty much any other vantage point (besides that of a student blogging about homological algebra, having already taken algebraic topology) we are doing everything backwards! Indeed, most often one studies algebraic topology and develops homological algebra in tandem, only as a tool.

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Regardless of why we are considering singular homology, a more pressing question may be: what is it? The notion of homology arises from simple topological questions. For example, suppose that you are a student who has just completed a first course in general (point-set) topology. Having (mostly) completed books such as Kelley, Munkres, and Willard you feel relatively confident in your ability to answer most topological questions. Of course, asked if given space X is metrizable you easily answer: “Of course! It’s regular, Hausdorff, and has a countably locally finite basis!” But then, someone poses to you an innocuous enough looking question “Are there sphere \mathbb{S}^2 and the torus \mathbb{T}^2 homeomorphic?” You immediately suspect the answer is no, your intuition being that you can’t deform one into the other without tearing a new hole, or mending an already existing one. Luckily you have dealt with such problems many-a-time before, even having proved the non-homeomorphic nature of (ostensibly!) more difficult spaces. In fact, you have done this so many times before you have basically reduced such a question to a procedular science. You whip out your favorite point-set list of topological invariants, put on your reading glasses, and start comparing. Are they both compact–yes. Are they both connected–yes. Are they both path connected–yes. Do they enjoy exactly the same separation properties–yes, they’re both metrizable. Do they have the same number of cut points–yes. The list goes on, and on, and on. Eventually, to your astonishment, you have reached the end of your supreme list of invariants only to realize that \mathbb{S}^2 and \mathbb{T}^2 satisfy them all. It seems that this problem is not as easy as it first appeared.

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Stymied, you return to your original intuition–one space has “holes” and the other doesn’t. Unfortunately, whatever you mean by “holes” (in a rigorous sense) none of the properties on your list of invariants was sensitive enough to detect them. Determined to solve this problem (being the stubborn student of mathematics that you are) you decide that you are going to try to formalize the notion of holes, prove that “holieness” is a topological invariant, and then prove that the sphere and the torus have a different number of holes. Well, if one stops and thinks about it, that’s a pretty formidable task. Regardless, you forge ahead on your holy quest (pun intended).  You start playing around with your specific examples of the torus and the sphere. After about ten minutes you have come up with at least five different ways to detect holes–all of them incorrect. Why? Namely, all of them relied on thinking of the torus and sphere as being embedded in the ambient space \mathbb{R}^3 in which they live. For example, your first idea was the “I can put my finger through one” approach where you attempt to formalize the notion that a line can pass through one object, but not the other.

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Another ten minutes go by, all the while thinking that you need to create a way of detecting holes that “stays on the surface”. Being the brilliant student that you are, a truly remarkable idea comes to you. The idea came to you while doodling on your favorite surfaces (I’m starting to sound like Vi Hart). You notice that every time you drew a “triangle” on the sphere it was the “boundary” of some filled in triangle. That said, on the torus there are lots of “triangles” which intuitively can’t be the boundary of a filled in triangle, since the hole presents proper filling. Great, so you’re well on your way to making everything nice and formal–you have the base idea of how to measure holes, you just need to formalize it. This is where all your time reading math books and listening to people smarter than you has paid off, for an old mathematical mantra pops into your head “Math is hard, linear algebra is easy.” So, you attempt to phrase your problem in the easy world of linear algebra. Of course, being the algebraically minded student that you are, you realize that this should entail creating a functor \mathbf{Top}\to R\text{-}\mathbf{Mod} for some ring R.

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Letting the idea ferment for a while (say, a hundred years or so) you finally figure out how to go about doing all of this. You realize that if you take something, like a filled in triangle, then it’s boundary can be thought of as a “sum” of filled in objects of one lower dimension–lines. Moreover, this sum has the property that it starts where it begins. In fact, you start to realize that the existence of holes by finding objects which “should” be the boundary of some filled in objects, but aren’t, can be phrased entirely in terms of these “closed” objects and the boundaries of filled in objects. Indeed, a space should have holes if there are “more” closed objects than there are boundaries of filled in objects.

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February 20, 2012 Posted by | Algebraic Topology, Topology | , , , | 8 Comments