Singular Homology (Pt. I)
Point of Post: In this post we discuss the notion of the singular homology of a topological space.
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.
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 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 and the torus 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 and satisfy them all. It seems that this problem is not as easy as it first appeared.
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 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.
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 for some ring .
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.
Ok, so let’s start to create a dictionary between the above intuitive ideas and the formalities that make-up mathematics.
Probably the first place we need to start with is what we do we mean by “triangles”. The first problem is that triangles, in the sense that we are thinking of them, can only detect so-called “one-dimensional” holes. For example, it’s clear that the sphere and the torus both have another type of hole, the one which is created by their hollowness. That said, the triangles cannot pick this up because it is a fundamentally two-dimensional phenomenon. That said, as we shall see, if we had instead started by thinking about the analogous situation, but looking at the boundaries of filled in tetrahedrons this can detect such holes. Thus, if we want to detect holes of all dimensions we cannot limit ourselves to just filled in triangles, or even filled in tetrahedrons, but their analogies in every dimension. To this end we quickly discuss the notion of the standard -simplex.
Recall that a subset is convex if for every two points the line-segment joining the two is contained in . It’s easy to see that the intersection of any number of convex subsets of is convex and so given we can define the convex hull of , denoted or if , to be the intersection of all convex supersets of . We then define the standard -simplex , denoted , to be the convex set where are the standard basis vectors in (i.e. , etc.). We see then that is a point, is a line, is a filled in triangle, is a filled in tetrahedron, etc.
Of course, the notion of an “-simplex” on a general topological space (even things as tame as surfaces!) is not well-defined. That said, we are intuitively trying to “put -simplexes” on our space, in a way made slightly more apparent by imagining that one gets our notion spherical triangles by “putting a triangle” on the sphere. Of course, doing topology, putting something into a space should correspond to mapping (continuously of course) an object into our space. Thus, for us an -simplex “on ” shall mean a continuous map , called a singular -simplex.
Remark: The quantifier “singular” is there to denote that there can be some collapsing of the triangle–the mapping needn’t be faithful in any substantive way. For example, may be a constant map.
 Rotman, Joseph J. An Introduction to Algebraic Topology. New York: Springer-Verlag, 1988. Print.
 Bredon, Glen E. Topology and Geometry. New York: Springer-Verlag, 1993. Print.
 Hatcher, Allen. Algebraic Topology. Cambridge: Cambridge UP, 2002. Print.