Stephen Paul King (firstname.lastname@example.org)
Wed, 17 Nov 1999 15:16:00 -0500
On 16 Nov 1999 18:00:01 -0600, in sci.physics.research
email@example.com (John Baez) wrote:
>In article <firstname.lastname@example.org>, <email@example.com> wrote:
>>Thanks for people's response to my question about fibre bundles.
>>I have another along the same lines. What is a cohomology (or
>>cohomology theory) and how is it used in physics?
>I forget how much math you know, so I'll start out by saying some
>incredibly elementary stuff and then shoot forwards rapidly to some
>more sophisticated stuff. If you know a medium amount of math,
>you'll probably be bored at first and then completely confused.
>Okay? So: I'll concentrate on what cohomology theory *is* and
>leave the physics applications for someone else.
>If you have a space, it can have holes of various different
>dimensions. A cohomology theory is a way of defining what
>you mean by holes, and it lets you add and subtract these holes.
>First you gotta understand how we keep track of the dimension of
>a hole! It takes a bit of getting used to, so don't argue, just
>pay close attention:
>Consider a doughnut. We say this has a 1-dimensional hole because
>it's possible to put a circle in a doughnut which encircles the
>hole of the doughnut, and we can't "pull this circle tight" -
>contract it to a point - while keeping it in the doughnut.
>Since a circle is 1-dimensional we say the hole is 1-dimensional.
>A doughnut has no holes of dimensions other than 1.
>Now consider a basketball: just the rubber skin, not the air inside.
>We say this has a 2-dimensional hole because it's possible to put a
>sphere inside the skin of the basketball which wraps around the
>air inside, and we can't "pull this sphere tight" - contract it
>to a point - while keeping it in the skin of the basketball.
>Since a sphere is 2-dimensional we say the whole is 2-dimensional.
>A basketball has no holes of dimensions other than 2.
>Now consider an inner tube: just the rubber skin, not the air inside.
>This is like a hollow doughnut! It's more complicated than the previous
>examples because it has more than one kind of hole. It has a 1-dimensional
>hole corresponding to how you can wrap a circle around it the long way,
>just like you can with the doughnut. But it also has a 1-dimensional
>hole corresponding to how you can wrap a circle around it the short way!
>It also has a 2-dimensional hole corresponding to how you can put a
>torus inside the skin of the inner tube. This 2-dimensional hole
>is a bit like the example of the basketball, except now we're using a
>torus to wrap around the air inside the inner tube, instead of a sphere.
>If you're smart you can figure out other ways to wrap a circle around
>the inner tube: it can sort of *spiral* around the inner tube. But this
>kind of 1-dimensional hole is a linear combination of the 1-dimensional
>holes I already discussed. If the circle spirals n times around the
>short way while it spirals m times around the long way, we say the hole
>it represents is nx + my, where x and y form the "basis" of 1-dimensional
>holes described in the previous case.
>Okay: in general, the cohomology of a space X is a bunch of abelian
>groups H^n(X), one for each integer n. The idea is that H^n(X) consists
>of all the n-dimensional holes, and we can add and subtract these holes.
>Now, so far I've been describing holes by looking at *manifolds* mapped
>into our space X. But we could use other things. It's actually very
>customary to use *simplices*. Different ways give different cohomology
>theories. If we use simplices we get something called "singular cohomology
>theory" or "ordinary cohomology theory". If we use manifolds we get
>something called "cobordism theory". There are actually lots of different
>kinds of cobordism theory depending on what sort of manifolds we use:
>there's "oriented cobordism theory" and "unoriented cobordism theory"
>and "smooth cobordism theory" and "piecewise-linear cobordism theory"
>and so on. And there are lots of other cohomology theories, too, like
>K-theory and stable homotopy theory and so on. These cohomology theories
>are related in lots of useful ways.
>In physics, perhaps the most popular cohomology theory is "deRham
>cohomology theory". Here we use differential forms to keep track
>of holes in a smooth manifold. Since differential forms are very
>important in physics, it's not surprising that deRham theory has lots
>of interesting applications. Stokes' theorem, Gauss' theorem, Green's
>theorem - they're all just the tip of the iceberg called deRham
>theory. You can find an elementary introduction to deRham cohomology
>in my book "Gauge Fields, Knots and Gravity", and in lots of other places,
>like Flanders' book "Differential Forms with Applications to the Physical
>Sciences", or von Westenholz' book "Differential Forms in Mathematical
>Physics". If you're a physicist, deRham theory is the place to start!
>Okay, finally for some more high-powered stuff. For every cohomology
>theory H^n, there's a bunch of spaces S(n) called the "classifying spaces"
>of that cohomology theory. The cohomology group H^n(X) of any space
>X is really just given by [X,S(n)] - the set of homotopy classes of maps
>from X to S(n), which turns out to be a group, thanks to some special
>stuff about S(n). For example, for ordinary cohomology the classifying
>space is called an "Eilenberg-MacLane space" and denoted K(Z,n). In
>general, for any cohomology theory, the spaces S(n) fit together into a
>gadget called a "spectrum". So in a sense, cohomology theory is really
>the study of spectra! This is how the experts think about it. But to
>really understand this viewpoint, you gotta understand what's so great
>about spectra. Ultimately it turns out that spectra are just a super-duper-
>generalization of abelian groups: they are "infinitely categorified,
>infinitely stabilized abelian groups". This is why they're so great.
>Anyone who wants to learn more about this should read Adams' book "Infinite
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