From: rusin@washington.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Alg. Topology texts -- RECOMMENDATIONS??? Date: 2 Nov 1994 05:56:05 GMT Regarding the question "what is a homology group": In article , Gordon McLean Jr. wrote: >john baez (baez@guitar.ucr.edu) wrote: >: ... nth homology >: group with real coefficients is a vector space whose dimension is the >: number of "n-dimensional holes" in the space you've got. > >Can you say a little more about what an "n-dimensional hole" is? > >Hopefully the answer is not something that just boils down to, "Well, >it's an n-cycle of the space, which is not the boundary of any >(n + 1)-chain of the space" :-) I must agree with the last poster. The concept of "n-dimensional holes" is fine until you need to look a bit more subtly and, for example, distinguish homology and homotopy groups. The classic example is the 2-dimensional torus S^1 x S^1, for which H_2 = Z but "there are no 2-d holes" (that is, \pi_2 is zero, as indeed are all higher homotopy groups). If it is any consolation, the (co)homology groups are representable functors, so that they count the homotopy classes of maps between the space X and certain other spaces (depending on the dimension n and the coefficient group G) -- namely the Eilenberg-MacLane spaces K(G, n). Inasmuch as these are geometrically rather complex spaces in general it does rather beg the question, "just what do the homology groups measure?" dave PS - don't even _think_ about enquiring about cohomology operations like the Steenrod square :-) ============================================================================== From: baez@guitar.ucr.edu (john baez) Newsgroups: sci.math Subject: Re: Alg. Topology texts -- RECOMMENDATIONS??? Date: 2 Nov 1994 21:17:49 GMT In article <39798b$oi8@mp.cs.niu.edu> rusin@washington.math.niu.edu (Dave Rusin) writes: >I must agree with the last poster. The concept of "n-dimensional holes" is >fine until you need to look a bit more subtly and, for example, >distinguish homology and homotopy groups. The classic example is >the 2-dimensional torus S^1 x S^1, for which H_2 = Z but "there >are no 2-d holes" (that is, \pi_2 is zero, as indeed are all higher homotopy >groups). Well, there's more to holes than round holes. :-) See my post on the hole in the doughnut... that is, the hole in the *hollow* doughnut. >If it is any consolation, the (co)homology groups are representable functors, >so that they count the homotopy classes of maps between the space X >and certain other spaces (depending on the dimension n and the coefficient >group G) -- namely the Eilenberg-MacLane spaces K(G, n). Inasmuch as these >are geometrically rather complex spaces in general it does rather beg >the question, "just what do the homology groups measure?" I *don't* think it's good to toss ones geometrical intuition out the window when doing algebraic topology, even though things get hairy at times and one must exert a fair amount of care. In particular, what I'm trying to do here is instill a healthy lack of awe for the techniques of algebraic topology. When experts start talking about "representable functor" and "Eilenberg-MacLane space", us mere mortals tend to get the impression that algebraic topology is infinitely beyond our ken. In fact, of course, it's not really that bad. Once you understand something in mathematics, at least if it's important and fundamental, it's usually pretty simple. For example, even though K(G,n)'s are sort of nasty, you can still get a certain grip on them by saying to yourself: okay, I want to get myself a space that has pi_n = G and all the other homotopy groups zero. So what do I do? I want to get myself a bunch of n-dimensional holes that give me pi_n = G. These are *round* holes now, since we're talking homotopy, not homology. So I take a bunch of S^n's, one for each generator of G, and wedge them all together. That means we have a bunch of n-dimensional balloons on strings, one for each generator of G, with the ends of all the strings tied together. This has pi_n equal to a free group on n generators. Then we want to impose the relations in the group G. So we take one n+1-dimensional ball for each relation and glue it onto the space we've got in such a way as to kill off the relations. If I could draw better on ASCII I could explain better how you do this. I did an easy example of this in the case I treated in my previous post, where I was trying to get a space with pi_1 = Z_5 (hence H_1 = Z_5, which is what McLean wanted). That space, which I described as a quotient space of the disc, was really the result of taking a circle and glomming a disc onto it in such a way that the edge of the disc wrapped around the circle 5 times. So what we started with was an S^1, which has pi_1 equal to the free group on one generator, and we wanted to kill off 5 times that generator (to impose the relations we need to get Z_5), so we glued on a 2-ball that wrapped around 5 times. So if we do this trick, we get a space that has pi_1 = G, and it doesn't have any nonzero homotopy groups *below* dimension n, but it still might have some *above* dimension n, so we need to keep glomming on higher- and higher-dimensional balls to kill off the higher homotopy groups. This goes on forever but mathematicians have plenty of time for such activities. We get K(G,n). As Rusin points out, K(G,n) is cool because it's a space that knows all about n-dimensional homology with coefficients in the group G. To figure out H_n(X,G) for any space X, we just form the set of homotopy equivalence classes of maps from X to K(G,n), usually written [X,K(G,n)]. This turns out to be H_n(X,G). Whenever we have some invariant H(X) of topological spaces, and there is some "master space" K such that to compute H(X) for any space we just calculate [X,K], we say our invariant is a "representable functor" and that K "represents" X. A devilishly sneaky trick, no? It means that in some mystical sense our invariant H of topological spaces sort of "is" a topological space itself, the space K! By this I mean that K is "maximally complicated" when it comes to the invariant H. That's why Eilenberg-MacLane spaces sort of *have* to be complicated, except in simple cases. There is a nice branch of algebraic topology called generalized homology theory that's all about this. But one should start out with simple stuff, like holes. :-) ============================================================================== To: baez@math.ucr.edu (john baez) Subject: Re: Alg. Topology texts -- RECOMMENDATIONS??? From: rusin@math.niu.edu (Dave Rusin) Date: Wed, 2 Nov 94 16:41:15 GMT >In article gordon@atria.com (Gordon McLean Jr.) writes: >>Can you say a little more about what an "n-dimensional hole" is? > >Well of course the *precise* definition is the one you've given, but the >point is, you are supposed to think of it as what you get when you >poke a hole in R^{n+1}. In other words, it's roughly the sort of thing >an n-dimensional surface could get caught on. An "n-cycle" is just a more >sophisticated incarnation of the concept of an n-dimensional surface. > >If you think of it this way it's supposed to obvious for example that >the 2-dimensional torus has H_2 = Z, because it's what you get when you >hollow out a doughnut --- the "hollowing out" carves out a 2-dimensional >hole. And it should also be obvious, perhaps a little less so, that the Be careful here -- it almost sounds as if your interpretation of homology is dependent upon its embedding in R^3. If you view the torus as a product (in R^4) of two circles in R^2, then it's not so clear where the "hollowing out" occurs -- you'd be looking at a codimension-2 submanifold. But the idea of "getting caught on" is really quite good. Indeed, Poincare duality "interprets" each homology class as a class in a different homology group; the pairing is via the cap product. This thing not only looks like an upside-down cup product, it looks like an intersection symbol. And indeed, this is how it arose historically. You intersect codimension k stuff with codimension l stuff and get (assuming general position) codimension k+l. Personally I have never been able to keep this vision of classes from squirming around uncontrollably, but it is a historically vindicated one. >Well, if I am not mistaken, the way you get a space with H_1 = Z_n is >pretty similar. You just take a disc and identify all points theta on >the unit circle (where theta goes from 0 to 2pi) with the points >theta + 2pi/n. That way, the path from 0 to 2pi/n is a noncontractible >loop, but n times it is contractible (and use the relation between pi_1 >and H_1). You are not mistaken. Another way to see this is you write Z_n as Z/nZ= (one generator, one relation). You create a space with this generator by simply taking a circle. Then you add a relation by gluing on a disc in the prescribed way, that is, let X = Disc u S^1 modulo the equivalence relation that says exp(theta i) in the disc is to be identified with exp(theta n i) in the circle. More generally, you can get a space with any fundamental group in this way by simply constructing a 2-dimensional CW complex with 1-cells for each generator and 2-cells for each relation. When the fundamental group is abelian, this is also H_1. >funny business. By the way, the usual term for "gluing together at one >point" is "wedging". So you get spaces with any desired homology groups Careful here. The wedge product is the coproduct in the category of pointed spaces (spaces with one point declared to be a "basepoint") This is simply the coproduct in the category of spaces (or sets), namely disjoint union, modulo an equivalence relation that declares the two basepoints equivalent. "gluing" is a more general term that involves three spaces X, Y, and A together with maps A-> X and A-> Y (typically, one of these is just an inclusion of a subspace); one forms the disjoint union again (X U Y) and then takes a quotient (XUY)/~ where the equivalence relation is set by declaring the images of each a in A (both in X and in Y) to be equivalent. The wedge product is the special case A=point. Incidentally, in the category of pointed spaces, suspension is not (XxI)/(Xx1, Xx0) but rather (XxI)/(Xx1 U Xx0 U *xI), where * is the basepoint of X. This seems messy, and it is; it's what you have to do in homotopy theory, but in homology theory the unpointed spaces are OK. >Now there is more to life than getting the homology groups right but at >least that's a start! Yeah, unfortunately you can get all the homology groups right and not have the right space. For example, the wedge product of the figure 8 and the sphere S^2 has the same homology groups as the torus, but the two spaces aren't even homotopy equivalent. Among other things, the ring structures in cohomology will not be the same. FWIW, it is true that IF two spaces X and Y have the same HOMOTOPY groups, and IF there is a map X->Y which INDUCES the isomorphism, and IF both X and Y are CW complexes, then X and Y have at least the same HOMOTOPY TYPE. (You'll never get anything like "homeomorphic" using only functors from the homotopy category). ============================================================================== From: baez@math.ucr.edu (john baez) Subject: Re: Alg. Topology texts -- RECOMMENDATIONS??? To: rusin@math.niu.edu (Dave Rusin) Date: Wed, 2 Nov 94 16:41:15 GMT > John, you have a pretty good intuition about homology, but I thought I'd > help you pin things down a bit Thanks for the various corrections, expansions, etc.. Actually, since McLean seemed familiar with the formal stuff and really lacking in intuition, I was trying to go heavily in the other direction and be extremely geometrical and touchy-feely. [deletia - djr] Best, John [above letter djr -> baez was attached, now deleted -- djr] ============================================================================== From: vidynath@math.ohio-state.edu (Vidhyanath Rao) Newsgroups: sci.math Subject: Re: Alg. Topology texts -- RECOMMENDATIONS??? Date: 3 Nov 1994 08:48:37 -0500 The trouble with simple geometric approaches to homology is that the end product can be quite complicated. I will try to approach the problem based on the history as I understand it. The basic idea is simple. We want to know in how many "different" ways "geometric objects" in our space can fail to be the boundaries of other "geometric objects" of the same type. Emmy Noether (sp?) is credited with the idea of organizing this information into a (graded) abelian group. This seems obvious today, so I will assume that it is a given :-) Thus we want to be able to form formal linear combinations (may be with some geometric meaning attached to -1 as in path integrals). Next thing to decide is what our "geometric objects" should be. The apparrently simplest would be embedded/immersed manifolds. However, this runs into serious problems when we try to compute. Another possibility for our "geometric objects" are embedded convex polyhedra. This is a bit better, but still difficult when we try to relate different polyhedra. So we go whole hog and consider only the simplices of some triangulation. This turns out to be computable and, even more strangely, useful. But it works only for spaces that can be triangulated. Our geometric objects are not so geometric anymore. But who cares. We got a good thing going. Now we want to define homology for any space. There are two ways in which to do this: Approximate our space by "good spaces" or redefine what geometric objects are. The first eventually leads to "Steenrod homology" (a better version of Cech homology). Of the various approaches to the second, singular homology proved to be most tractable. Our geometric objects are continuous images of polyhedra (which are of course linear combination of simplices). Polyhedra have sharp corners, but we like smooth curves (male chaunism?). Taking the lesson from singular theory, we ask "can we use continuous images of manifolds instead of polyhedra?", as Steenrod did. The short answer is "No, not if you want a nice simple theory". The long answer is that you get a generlized homology theory: excision, exactness and homotopy invariance work, but the homology of a point is enormous. These go by the name of bordism theories. Depending on what kind of manifolds we allow, we get different theories. It turns out that we can still get our old friend the singular theory if we allow "singularities". Doing this properly is such a mess that few actually read the stuff. (the idea was proposed, as far as know, by Sullivan. The usual reference is a paper by N.Baas in Math. Scand. 197?). Another way to do this is by taking the suitable dual to harmonic differential forms. Done correctly, this does lead to a theory equivalent to singular theory, but the cycles may have singularities (but not too wild). This is usually considered part of geometric measure theory which I don't understand. Now it should be clear why visualizing homology is difficult: We want to visualize immersed manifolds. But the resulting theory is too messy. For "nice Riemannian manifolds" we can get away with this, if we know what equivalence relation to put on the cycles. In particular, this works great for surfaces and most 3-manifolds. But in general this does not work. Which is why we need messy definitions. Anyway, this is what I think of as an 'n-dimensional hole": An n-dimensional "manifold" in the space that ought to bound something (does bound something somewhere else), but doesn't. Just don't ask me what "manifold" is (as opposed to manifold without quotes). ============================================================================== From: vidynath@math.ohio-state.edu (Vidhyanath Rao) Newsgroups: sci.math Subject: Re: Alg. Topology texts -- RECOMMENDATIONS??? Date: 4 Nov 1994 16:39:02 -0500 I forgot to include any references. Here are some: E. Betti: Sopra gli spazi di un numero qualunque di dimensioni Ann Math Pura Appl 4(1871) Defined Betti numbers using submanifolds that do not bound. H. Poincare: See his series of papers on Analysis Situs. Started off using Betti's definition. Switched over to simplicial chains to make things more rigorous. R. Stong: Notes on cobordism theory (Princeton University Press, 1968) Standard reference for bordism. S. Buoncristiano, C. P. Rourke and B. J. Sanderson: A geometric approach to homology theory (Cambridge University Press 1976). Any homology theory can be defined as a "bordism".