**Stephen Paul King** (*stephenk1@home.com*)

*Wed, 17 Nov 1999 15:16:00 -0500*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**koichiro matsuno/7129: "[time 1003] Re: [time 990] Re: [time 987] a fundamental question on QM time"**Previous message:**Stephen Paul King: "[time 1001] The Logic of Time"

On 16 Nov 1999 18:00:01 -0600, in sci.physics.research

baez@galaxy.ucr.edu (John Baez) wrote:

*>In article <80rr9p$1f7$1@nnrp1.deja.com>, <tedsung6674@my-deja.com> wrote:
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*>
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*>>Thanks for people's response to my question about fibre bundles.
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*>>I have another along the same lines. What is a cohomology (or
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*>>cohomology theory) and how is it used in physics?
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*>
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*>I forget how much math you know, so I'll start out by saying some
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*>incredibly elementary stuff and then shoot forwards rapidly to some
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*>more sophisticated stuff. If you know a medium amount of math,
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*>you'll probably be bored at first and then completely confused.
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*>Okay? So: I'll concentrate on what cohomology theory *is* and
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*>leave the physics applications for someone else.
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*>
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*>If you have a space, it can have holes of various different
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*>dimensions. A cohomology theory is a way of defining what
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*>you mean by holes, and it lets you add and subtract these holes.
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*>
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*>First you gotta understand how we keep track of the dimension of
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*>a hole! It takes a bit of getting used to, so don't argue, just
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*>pay close attention:
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*>
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*>Consider a doughnut. We say this has a 1-dimensional hole because
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*>it's possible to put a circle in a doughnut which encircles the
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*>hole of the doughnut, and we can't "pull this circle tight" -
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*>contract it to a point - while keeping it in the doughnut.
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*>Since a circle is 1-dimensional we say the hole is 1-dimensional.
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*>A doughnut has no holes of dimensions other than 1.
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*>
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*>Now consider a basketball: just the rubber skin, not the air inside.
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*>We say this has a 2-dimensional hole because it's possible to put a
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*>sphere inside the skin of the basketball which wraps around the
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*>air inside, and we can't "pull this sphere tight" - contract it
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*>to a point - while keeping it in the skin of the basketball.
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*>Since a sphere is 2-dimensional we say the whole is 2-dimensional.
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*>A basketball has no holes of dimensions other than 2.
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*>
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*>Now consider an inner tube: just the rubber skin, not the air inside.
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*>This is like a hollow doughnut! It's more complicated than the previous
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*>examples because it has more than one kind of hole. It has a 1-dimensional
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*>hole corresponding to how you can wrap a circle around it the long way,
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*>just like you can with the doughnut. But it also has a 1-dimensional
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*>hole corresponding to how you can wrap a circle around it the short way!
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*>It also has a 2-dimensional hole corresponding to how you can put a
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*>torus inside the skin of the inner tube. This 2-dimensional hole
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*>is a bit like the example of the basketball, except now we're using a
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*>torus to wrap around the air inside the inner tube, instead of a sphere.
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*>
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*>If you're smart you can figure out other ways to wrap a circle around
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*>the inner tube: it can sort of *spiral* around the inner tube. But this
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*>kind of 1-dimensional hole is a linear combination of the 1-dimensional
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*>holes I already discussed. If the circle spirals n times around the
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*>short way while it spirals m times around the long way, we say the hole
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*>it represents is nx + my, where x and y form the "basis" of 1-dimensional
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*>holes described in the previous case.
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*>
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*>Okay: in general, the cohomology of a space X is a bunch of abelian
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*>groups H^n(X), one for each integer n. The idea is that H^n(X) consists
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*>of all the n-dimensional holes, and we can add and subtract these holes.
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*>
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*>Now, so far I've been describing holes by looking at *manifolds* mapped
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*>into our space X. But we could use other things. It's actually very
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*>customary to use *simplices*. Different ways give different cohomology
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*>theories. If we use simplices we get something called "singular cohomology
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*>theory" or "ordinary cohomology theory". If we use manifolds we get
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*>something called "cobordism theory". There are actually lots of different
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*>kinds of cobordism theory depending on what sort of manifolds we use:
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*>there's "oriented cobordism theory" and "unoriented cobordism theory"
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*>and "smooth cobordism theory" and "piecewise-linear cobordism theory"
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*>and so on. And there are lots of other cohomology theories, too, like
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*>K-theory and stable homotopy theory and so on. These cohomology theories
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*>are related in lots of useful ways.
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*>
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*>In physics, perhaps the most popular cohomology theory is "deRham
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*>cohomology theory". Here we use differential forms to keep track
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*>of holes in a smooth manifold. Since differential forms are very
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*>important in physics, it's not surprising that deRham theory has lots
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*>of interesting applications. Stokes' theorem, Gauss' theorem, Green's
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*>theorem - they're all just the tip of the iceberg called deRham
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*>theory. You can find an elementary introduction to deRham cohomology
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*>in my book "Gauge Fields, Knots and Gravity", and in lots of other places,
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*>like Flanders' book "Differential Forms with Applications to the Physical
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*>Sciences", or von Westenholz' book "Differential Forms in Mathematical
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*>Physics". If you're a physicist, deRham theory is the place to start!
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*>
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*>Okay, finally for some more high-powered stuff. For every cohomology
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*>theory H^n, there's a bunch of spaces S(n) called the "classifying spaces"
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*>of that cohomology theory. The cohomology group H^n(X) of any space
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*>X is really just given by [X,S(n)] - the set of homotopy classes of maps
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*>from X to S(n), which turns out to be a group, thanks to some special
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*>stuff about S(n). For example, for ordinary cohomology the classifying
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*>space is called an "Eilenberg-MacLane space" and denoted K(Z,n). In
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*>general, for any cohomology theory, the spaces S(n) fit together into a
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*>gadget called a "spectrum". So in a sense, cohomology theory is really
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*>the study of spectra! This is how the experts think about it. But to
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*>really understand this viewpoint, you gotta understand what's so great
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*>about spectra. Ultimately it turns out that spectra are just a super-duper-
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*>generalization of abelian groups: they are "infinitely categorified,
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*>infinitely stabilized abelian groups". This is why they're so great.
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*>Anyone who wants to learn more about this should read Adams' book "Infinite
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*>Loop Spaces".
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*>
*

**Next message:**koichiro matsuno/7129: "[time 1003] Re: [time 990] Re: [time 987] a fundamental question on QM time"**Previous message:**Stephen Paul King: "[time 1001] The Logic of Time"

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