The Algebraic Topology Chain Gang
Copyright © 2005, David A. Epstein.
All Rights Reserved.

    "Say it ain't so. Don't tell me they're going to torture us with that homology theory again." Jetsonsky was panicking.
    "Of course not. This time, they're planning to bombard us with cohomology." I just couldn't resist needling him.
    "What? That's even worse. How could they do that to us."
    "Pure evil, that's what it is. Pure evil."
    "Are they that heartless? Is that it?"
    "Certainly not. They have hearts, big hearts. It's just that they're filled with, Pure Evil."
    Jetsonsky didn't just give me an incredulous look. For a moment, I thought he hated me as well. He undoubtedly was confusing the messenger with the perpetrator. In any case, it became academic as the Man walked up to the podium and started his lecture.
    "And now class, we are going to discuss relative homology groups. Let's start with our good friend, the abstract simplicial complex K, and a closed subcomplex L contained in K. The chain group, Cp(L) is a subgroup of Cp(K). We define a relative chain group of K modulo L as the difference groups: Cp(K/L) = Cp(K) - Cp(L). ..."
    We rattled our Shackles in unison, trying to make a political statement without infuriating our captors. We just wanted to get their attention and take a stand; but our "teacher" was ignoring us, at least that's the attitude he conveyed.
    "So, the relative homology groups of K mod L are Hp(K/L) = Zp(K/L) - Bp(K/L). For a relative chain to be a relative cycle, the boundary operator delta-Cp must equal 0. This means the boundary surrounding you pathetic creatures must equal zero, and that, of course, means it will crush all of you to death. And now, we will prove this proposition, which will unquestionably lead to your demise."
    "NO, No, please don't start that proof," cried out Jetsonsky.
    The "professor" started writing the proof on the board.
    "No, STOP IT. DON'T DO IT."
    "Err, OK. I'll leave it as a homework exercise. You should have fun with it. And besides, you didn't actually believe your surrounding boundary will equal zero, now, did you? What gullible peons you are. The boundary will simply lie in L, silly ones. Ha. Hopelessly chained to this abstract complex."
    We realized it was going to be a troublesome period for us. Our chains were very tight. There was no obvious escape. The professor continued with his tirade: "Now, we will show that the homology sequence of a pair (K,L) is exact." He showed the proof on the board. With each step of the proof, we felt needles passing right through us, as if we were subjected to some type of topological voodoo. It was excruciatingly painful. We desperately tried to break free from our enslavement, but with every attempt to gain our freedom, we encountered an even stronger resistance to keep us chained together.
    "OK, now we'll demonstrate the homomorphisms of exact sequences. For two exact sequences, call them (Gi, Wi) and (Hi, Oi), there is a set of homomorphisms J = (Ji), where Ji:Gi-->Hi, which itself is a homomorphism mapping the first sequence to the second, if Ji-1*Wi = Oi*Ji."
    "Please, enough of this already," Jetsonsky cried out. We were feeling squeezed by it all. It was apparently just too much for Jetsonsky to handle.
    "Oh, but it's delightful to think about, don't you think? I believe you would find it enjoyable if you could just relax. You see, it's actually these homomorphisms, that you're so eager to learn about, that are confining you between two encrypted sequences. It's certainly fascinating, wouldn't you agree? You can only escape if you know how to decrypt them, but of course you'll never figure that out. Ooooh yeah, that's so true."
    Perhaps he was right. This could very well be an unsolvable problem. To state that the proofs were overwhelming would be a significant understatement. Step by step, we felt the torment of trying to comprehend such mind-numbing material. We could never imaging living through such rigorous torture in our wildest dreams. I certainly thought about the day we would escape this topological prison and tell the world about our struggles. Would Amnesty International take up our cause and fight for our release? The major problem they would be facing is most people would not believe we were encapsulated in cohomological spaces and actually not enjoying it! It defied what people have been conditioned to believe, namely that it was the highest form of leisure known to humankind. Why, so many people living adventurous alternative lifestyles have done everything from going on homological excursions to colonizing such realms for prolonged periods of time. A plethora of favorable stories about the subject have even surfaced in the popular literature. But I digress.
    Our fate was undoubtedly sealed.
    "OK, anybody care for their daily dose of you-know-what? Silly me; of course you all care. Now it's time for the Excision Theorem, kiddies, or maybe it's more appropriate to call it your Excise Tax. I'm sure your faithful tax authorities will appreciate that one ... Alright, enough with the small talk. Back to work ... If we have L being a closed subcomplex of K, then K-L is an open subcomplex of K. Pretty simple, eh? Well, wait till you see the proof. That'll really kill you." And on and on it went. I can't discuss what occurred after that, for it would be too graphic for the children in our audience.
    Our captors did give us a break, though it was for an insignificant period of time. We hardly had time to recuperate before they were assaulting us with their next round of ammunition. "Continuing on with our discussion of the Excision Principle, let's take our beloved K and L complex and subcomplex respectively. And you remember M, right? That's our open subcomplex of K. We are going to prove that the injection mapping of K-M into K induces an isomorphism of Hp[(K-M)(L-M)] onto Hp(K/L) for each dimension p."
    There was a force pressing against me. My surrounding space was folding up. It appeared that my prison cell was being configured into the shape of a fortune cookie, and I was destined to be the fortune, but I was not feeling the least bit fortunate in any shape or form. The sole redeeming aspect, at that instance anyway, were the beams of light that trickled through the visible slit above; but within a couple of minutes, the encapsulating structure was flattened out, and the light beams no longer appeared.
I was immersed in total darkness and I felt compressed. Was I submerged in one of those complexes the professor was describing? Perhaps I was being isomorphisized, so to speak? Whatever it was, I was isolated from the others in the "classroom". It was an eerie feeling which lasted until ...
    Within a few minutes, the area filled with light. A few of the "students" slid down an S-shape slide and entered the space. That was reassuring in a peculiar way, for I now had fear and anxiety to share with the others. We looked at each other, wondering what this was all about. The expressions on their faces most clearly conveyed perplexity and disorientation. What gave us some hope is we were no longer chained together. We experienced some sense of liberation, crammed as we were in this enclosed space.
    "Whatever this is, uncomfortable as we are about the unknown, it's certainly better than where we were." I wasn't too confident about my utterance; I just wanted to exude a sense of optimism about our current situation.
    "We must have undergone some strange type of conformal mapping," said Jetsonsky.
    "More likely, it was a homomorphic transformation," chimed in another cohort.
    "Who knows. All we can say for sure is we're liberated from our chains, but confined to this abstract space," said another.
    "No, class, you're all wrong," said a voice emanating from above.
    "It can't be," said
    "Oh, but it is," replied the voice.
    "You mean, ..." repeated the second cohort.
    "That's right. It's GOD almighty," boomed the unseen voice in bassy, resonating tones.
    "Holy ..."
    "Watch your tongue." The unseen voice emerged into full view, puncturing through our enfolded space. He was slowly descending on a swing, swaying back and forth. He was wearing a top hat, a cool pair of steel-rim sunglasses, and smoking a fat cigar. At first, I could not make out who it was. Once he was within a few feet of us, however, I recognized who it was.
    "Good grief. We thought we lost you," I said.
    "Ah, come on my children. You know very well I'll always be with you. We're family, we love each other, don't you see?! And we're on a perpetual journey, you might say, through transference to different spaces," said the professor, puffing on his cigar. "Yeah, that's it; it most certainly is. And now, it's time to continue with the lecture. Except this time, you're in a real world setting. So you better brace yourselves and hold on to your seats, because you're inside a warped cohomology ring. Ha ha ha."
    "A what? Where are we?" asked
the second cohort.
    "What, are you hard of hearing as well as dumb?" replied the professor. Within a few seconds, we were reattached to some chains. Then, some force hurled us against one of the walls of the ring. A couple of my cohorts frantically tried to break free, but the only thing they managed to break, as they violently fell to the floor, were a few of their bones. "OK, let the non-klutzy types continue on. As you should be aware, the heart of algebraic topology is an algebraic structure attached to a topological space. And the cohomology ring is the algebraic structure attached to the complex K we've grown quite accustomed to. In this case, the structure is a series of cochains multiplied together. And all of you prisoners are fastened to the topological spaces known as manifolds. Got that?"
    He pressed a button on some remote control device. Within a few seconds, we were rapidly accelerating in our enclosed structures and being configured into various spatial arrangements. I felt dizzy as we zig-zagged in many different directions. Several of my fellow inmates slammed into the walls of these spatial prisons. It's as if there was a giant manipulating us like we were one large Rubik's Cube. This continued for a couple of minutes until we gradually came to a complete stop.
    "A lot of fun, wasn't it? One of our special amusement rides here at TopoloLand. Just a slight diversion before we get back to work; that's all it is." The professor wore a sardonic smile. He kept taking for a while before a giant white board appeared. W
hile he was speaking, he started writing on the board . "An important concept in cohomology is known as the cup product. For two cochains Cp & Cq, this cup product is Cp U Cq. It's the (p+q) cochain described by this formula:
    Cp U Cq((Vo * ... VpVp+1 ...Vp+q)) = Cp(Vo...Vp)*Cq(Vp ... Vp+q)"
    The more he spoke, the tighter are chains became. "Too much for you to handle, undoubtedly. Oh, my cup runneth over, you might say."

    The space surrounding us morphed into another shape. Strangely enough, it now appeared we were inside of a large cup. A few of us tried climbing the interior walls, but it was extremely slippery; everyone simply slid back to their starting points.
    "That homomorphism is keeping you secure in your cohomological ring prison; well, now it's a cuppedy cup." The professor was giving us a glaring look, as he peered down into the cup. Then he let out a loud cackle. This ignited our collective wrath. I exhorted the others to make a coordinated effort to bring him down. And bring him down we did. All of us slammed into the sides of the cup. We rattled him with continuous vibrations. It was the intensity of the motions that brought him down -- right into the bottom of the cup.
     We grabbed his gun, and shot a hole into the side of the cup.
    "That will enable us to invoke the cap product. This is the bridge between the cohomological and homological spaces, and that will allow us to get out of here. We'll escape to a homological space, of course." I now felt more assured in my pronouncements, even as the professor waved his fist and continued cursing us out in more ways that I could fathom.
    And escape we did, in chains and all.

Extracts from "Topology" by John G. Hocking and Gail S. Young; 1961 Dover.