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".