
There are some spooky facts in mathematics that you'd never guess in a million years... only when someone carefully works them out do they become clear. One of them is called "Bott periodicity".
A 0dimensional manifold is pretty dull: just a bunch of points. 1dimensional manifolds are not much more varied: the only possibilities are the circle and the line, and things you get by taking a union of a bunch of circles and lines. 2dimensional manifolds are more interesting, but still pretty tame: you've got your nholed tori, your projective plane, your Klein bottle, variations on these with extra handles, and some more related things if you allow your manifold to go on forever, like the plane, or the plane with a bunch of handles added (possibly infinitely many!), and so on.... You can classify all these things. 3dimensional manifolds are a lot more complicated: nobody knows how to classify them. 4dimensional manifolds are a lot more complicated: you can prove that it's impossible to classify them  that's called Markov's Theorem.
Now, you probably wouldn't have guessed that a lot of things start getting simpler when you get up around dimension 5. Not everything, just some things. You still can't classify manifolds in these high dimensions, but if you make a bunch of simplifying assumptions you sort of can, in ways that don't work in lower dimensions. Weird, huh? But that's another story. Bott periodicity is different. It says that when you get up to 8 dimensions, a bunch of things are a whole lot like in 0 dimensions! And when you get up to dimension 9, a bunch of things are a lot like they were in dimension 1. And so on  a bunch of stuff keeps repeating with period 8 as you climb the ladder of dimensions.
(Actually, I have this kooky theory that perhaps part of the reason topology reaches a certain peak of complexity in dimension 4 is that the number 4 is halfway between 0 and 8, topology being simplest in dimension 0. Maybe this is even why physics likes to be in 4 dimensions! But this is a whole other crazy digression and I will restrain myself here.)
Bott periodicity takes many guises, and I already described one in "week104". Let's start with the real numbers, and then throw in n square roots of 1, say e_{1},...,e_{n}. Let's make them "anticommute", so
e_{i} e_{j} =  e_{j} e_{i}
when i is different from j. What we get is called the "Clifford algebra" C_{n}. For example, when n = 1 we get the complex numbers, which we call C. When n = 2 we get the quaternions, which we call H, for Hamilton. When n = 3 we get... the octonions?? No, not the octonions, since we always demand that multiplication be associative! We get the algebra consisting of pairs of quaternions! We call that H + H. When n = 4 we get the algebra consisting of 2x2 matrices of quaternions! We call that H(2). And it goes on, like this:
C_{0} = R
C_{1} = C
C_{2} = H
C_{3} = H + H
C_{4} = H(2)
C_{5} = C(4)
C_{6} = R(8)
C_{7} = R(8) + R(8)
C_{8} = R(16)
Note that by the time we get to n = 8 we just have 16x16 matrices of real numbers. And that's how it keeps going: C_{n+8} is just 16x16 matrices of guys in C_{n}! That's Bott periodicity in its simplest form.
Actually right now I'm in Vienna, at the Schroedinger Institute, and one of the other people visiting is Andrzej Trautman, who gave a talk the other day on "Complex Structures in Physics", where he mentioned a nice way to remember the above table. Imagine the day is only 8 hours long, and draw a clock with 8 hours. Then label it like this:
0 R 7 1 R+R C 6 R H 2 C H+H 5 3 H 4The idea here is that as the dimension of space goes up, you go around the clock. One nice thing about the clock is that it has a reflection symmetry about the axis from 3 o'clock to 7 o'clock. To use the clock, you need to know that the dimension of the Clifford algebra doubles each time you go up a dimension. This lets you figure out, for example, that the Clifford algebra in 4 dimensions is not really H, but H(2), since the latter has dimension 16 = 2^{4}.
Now let's completely change the subject and talk about rotations in infinitedimensional space! What's a rotation in infinitedimensional space like? Well, let's start from the bottom and work our way up. You can't really rotate in 0dimensional space. In 1dimensional space you can't really rotate, you can only reflect things... but we will count reflections together with rotations, and say that the operations of multiplying by 1 or 1 count as "rotations" in 1dimensional space. In 2dimensional space we describe rotations by 2x2 matrices like
cos t sin t sin t cos tand since we're generously including reflections, also matrices like
cos t sin t sin t cos tThese are just the matrices whose columns are orthonormal vectors. In 3dimensional space we describe rotations by 3x3 matrices whose columns are orthonormal, and so on. In ndimensional space we call the set of nxn matrices with orthonormal columns the "orthogonal group" O(n).
Note that we can think of a rotation in 2 dimensions
cos t sin t sin t cos tas being a rotation in 3 dimensions if we just stick one more row and one column like this:
cos t sin t 0 sin t cos t 0 0 0 1This is just a rotation around the z axis. Using the same trick we can think of any rotation in n dimensions as a rotation in n+1 dimensions. So we can think of O(0) as sitting inside O(1), which sits inside O(2), which sits inside O(3), which sits inside O(4), and so on! Let's do that. Then let's just take the union of all these guys, and we get... O(∞)! This is the group of rotations, together with reflections, in infinite dimensions.
(Now if you know your math, or you read "week82", you'll realize that I didn't really change the subject, since the Clifford algebra C_{n} is really just a handy way to study rotations in n dimensions. But never mind.)
Now O(∞) is a very big group, but it elegantly summarizes a lot of information about rotations in all dimensions, so it's not surprising that topologists have studied it. One of the thing topologists do when studying a space is to work out its "homotopy groups". If you hand them a space X, and choose a point x in this space, they will work out all the topologically distinct ways you can stick an ndimensional sphere in this space, where we require that the north pole of the sphere be at x. This is what they are paid to do. We call the set of all such ways the homotopy group π_{n}(X). For a more precise description, try "week102"  but this will do for now.
So, what are the homotopy groups of O(∞)? Well, they start out looking like this:
n π_{n}(O(∞)) 0 Z/2 1 Z/2 2 0 3 Z 4 0 5 0 6 0 7 ZAnd then they repeat, modulo 8. Bott periodicity strikes again!
But what do they mean?
Well, luckily Jim Dolan has thought about this a lot. Discussing it repeatedly in the little cafe we tend to hang out at, we came up with the following story. Most of it is known to various people already, but it came as sort of a revelation to us.
The zeroth entry in the table is easy to understand. π_{0} keeps track of how many connected components your space has. The rotation group O(∞) has two connected components: the guys that are rotations, and the guys that are rotations followed by a reflection. So π_{0} of O(∞) is Z/2, the group with two elements. Actually this is also true for O(n) whenever n is higher enough, namely 1 or more. So the zeroth entry is all about "reflecting".
The first entry is a bit subtler but very important in physics. It means that there is a loop in O(∞) that you can't pull tight, but if you go around that loop twice, you trace out a loop that you can pull tight. In fact this is true for O(n) whenever n is 3 or more. This is how there can be spin1/2 particles when space is 3dimensional or higher. There are lots of nice tricks for seeing that this is true, which I hope the reader already knows and loves. In short, the first entry is all about "rotating 360 degrees and not getting back to where you started".
The second entry is zero.
The third entry is even subtler but also very important in modern physics. It means that the ways to stick a 3sphere into O(∞) are classified by the integers, Z. Actually this is true for O(n) whenever n is 5 or more. It's even true for all sorts of other groups, like all "compact simple groups". But can I summarize this entry in a snappy phrase like the previous nonzero entries? Not really. Actually a lot of applications of topology to quantum field theory rely on this π_{3} business. For example, it's the key to stuff like "instantons" in YangMills theory, which are in turn crucial for understanding how the πon gets its mass. It's also the basis of stuff like "ChernSimons theory" and "BF theory". Alas, all this takes a while to explain, but let's just say the third entry is about "topological field theory in 4 dimensions".
The fourth entry is zero.
The fifth entry is zero.
The sixth entry is zero.
The seventh entry is probably the most mysterious of all. From one point of view it is the subtlest, but from another point of view it is perfectly trivial. If we think of it as being about π_{7} it's very subtle: it says that the ways to stick a 7sphere into O(∞) are classified by the integers. (Actually this is true for O(n) whenever n is 7 or more.) But if we keep Bott periodicity in mind, there is another way to think of it: we can think of it as being about π_{1}, since 7 = 1 mod 8.
But wait a minute! Since when can we talk about π_{n} when n is negative?! What's a 1dimensional sphere, for example?
Well, the idea here is to use a trick. There is a space very related to O(∞), called kO. As with O(∞), the homotopy groups of this space repeat modulo 8. Moreover we have:
π_{n}(O(∞)) = π_{n+1}(kO)
Combining these facts, we see that the very subtle π_{7} of O(∞) is nothing but the very unsubtle π_{0} of kO, which just keeps track of how many connected components kO has.
But what is kO?
Hmm. The answer is very important and interesting, but it would take a while to explain, and I want to postpone doing it for a while, so I can get to the punchline. Let me just say that when we work it all out, we wind up seeing that the seventh entry in the table is all about dimension.
To summarize:
π_{0}(O(∞)) = Z/2 is about REFLECTING
π_{1}(O(∞)) = Z/2 is about ROTATING 360 DEGREES
π_{3}(O(∞)) = Z is about TOPOLOGICAL FIELD THEORY IN 4 DIMENSIONS
π_{7}(O(∞)) = Z is about DIMENSION
But wait! What do those numbers 0, 1, 3, and 7 remind you of?
Well, after I stared at them for a few weeks, they started to remind me of the numbers 1, 2, 4, and 8. And that immediately reminded me of the reals, the complexes, the quaternions, and the octonions!
And indeed, there is an obvious relationship. Let n be 1, 2, 4, or 8, and correspondingly let A stand for either the reals R, the complex numbers C, the quaternions H, or the octonions O. These guys are precisely all the "normed division algebras", meaning that the obvious sort of absolute value satisfies
xy = xy.
Thus if we take any guy x in A with x = 1, the operation of multiplying by x is lengthpreserving, so it's a reflection or rotation in A. This gives us a function from the unit sphere in A to O(n), or in other words from the (n1)sphere to O(n). We thus get nice elements of
π_{0}(O(1))
π_{1}(O(2))
π_{3}(O(4))
π_{7}(O(8))
which turn out to be precisely why these particular homotopy groups of O(∞) are nontrivial.
So now we have the following fancier chart:
π_{0}(O(∞)) is about REFLECTING and the REAL NUMBERS
π_{1}(O(∞)) is about ROTATING 360 DEGREES and the COMPLEX NUMBERS
π_{3}(O(∞)) is about TOPOLOGICAL FIELD THEORY IN 4 DIMENSIONS and the QUATERNIONS
π_{7}(O(∞)) is about DIMENSION and the OCTONIONS
Now this is pretty weird. It's not so surprising that reflections and the real numbers are related: after all, the only "rotations" in the real line are the reflections. That's sort of what 1 and 1 are all about. It's also not so surprising that rotations by 360 degrees are related to the complex numbers. That's sort of what the unit circle is all about. While far more subtle, it's also not so surprising that topological field theory in 4 dimensions is related to the quaternions. The shocking part is that something so basicsounding as "dimension" should be related to something so eruditesounding as the "octonions"!
But this is what Bott periodicity does, somehow: it wraps things around so the most complicated thing is also the least complicated.
That's more or less the end of what I have to say, except for some references and some remarks of a more technical nature.
Bott periodicity for O(∞) was first proved by Raoul Bott in 1959. Bott is a wonderful explainer of mathematics and one of the main driving forces behind applications of topology to physics, and a lot of his papers have now been collected in book form:
1) The Collected Papers of Raoul Bott, ed. R. D. MacPherson. Vol. 1: Topology and Lie Groups (the 1950s). Vol. 2: Differential Operators (the 1960s). Vol. 3: Foliations (the 1970s). Vol. 4: Mathematics Related to Physics (the 1980s). Birkhauser, Boston, 1994, 2355 pages total.
A good paper on the relation between O(∞) and Clifford algebras is:
2) M. F. Atiyah, R. Bott, and A. Shaπro, Clifford modules, Topology (3) 1964, 338.
For more stuff on division algebras and Bott periodicity try Dave Rusin's web page, especially his answer to "Q5. What's the question with the answer n = 1, 2, 4, or 8?"
3) Dave Rusin, Binary products, algebras, and division rings, http://www .math.niu.edu/~rusin/knownmath/95/division.alg
Let me briefly explain this kO business. The space kO is related to a simpler space called BO(∞) by means of the equation
kO = BO(∞) x Z,
so let me first describe BO(∞). For any topological group G you can cook up a space BG whose loop space is homotopy equivalent to G. In other words, the space of (basepointpreserving) maps from S^{1} to BG is homotopy equivalent to G. It follows that
π_{n}(G) = π_{n+1}(BG).
This space BG is called the classifying space of G because it has a principal Gbundle over it, and given any decent topological space X (say a CW complex) you can get all principal Gbundles over X (up to isomorphism) by taking a map f: X → BG and pulling back this principal Gbundle over BG. Moreover, homotopic maps to BG give isomorphic Gbundles over X this way.
Now a principal O(n)bundle is basically the same thing as an ndimensional real vector bundle  there are obvious ways to go back and forth between these concepts. A principal O(∞)bundle is thus very much like a real vector bundle of arbitrary dimension, but where we don't care about adding on arbitrarily many 1dimensional trivial bundles. If we take the collection of isomorphism classes of real vector bundles over X and decree two to be equivalent if they become isomorphic after adding on trivial bundles, we get something called KX, the "real Ktheory of X". It's not hard to see that this is a group. Taking what I've said and working a bit, it follows that
KX = [X, BO(∞)]
where the righthand side means "homotopy classes of maps from X to BO(∞)". If we take X to be S^{n+1}, we see
KS^{n+1} = π_{n+1}(BO(∞)) = π_{n}(O(∞))
It follows that we can get all elements of π_{n} of O(∞) from real vector bundles over S^{n+1}.
Of course, the above equations are true only for nonnegative n, since it doesn't make sense to talk about π_{1} of a space. However, to make Bott periodicity work out smoothly, it would be nice if we could pretend that
KS^{1} = π_{0}(BO(∞)) = π_{1}(O(∞)) = π_{7}(O(∞)) = Z
Alas, the equations don't make sense, and BO(∞) is connected, so we don't have π_{0}(BO(∞)) = Z. However, we can cook up a slightly improved space kO, which has
π_{n}(kO) = π_{n}(BO(∞))
when n > 0, but also has
π_{0}(kO) = Z
as desired. It's easy  we just let
kO = BO(∞) x Z.
So, let's use this instead of BO(∞) from now on.
Taking n = 0, we can think of S^{1} as RP^{1}, the real projective line, i.e. the space of 1dimensional real subspaces of R^{2}. This has a "canonical line bundle" over it, that is, a 1dimensional real vector bundle which to each point of RP^{1} assigns the 1dimensional subspace of R^{2} that is that point. This vector bundle over S^{1} gives the generator of KS^{1}, or in other words, π_{0}(O(∞)).
Taking n = 1, we can think of S^{2} as the "Riemann sphere", or in other words CP^{1}, the space of 1dimensional complex subspaces of C^{2}. This too has a "canonical line bundle" over it, which is a 1dimensional complex vector bundle, or 2dimensional real vector bundle. This bundle over S^{2} gives the generator of KS^{2}, or in other words, π_{1}(O(∞)).
Taking n = 3, we can think of S^{4} as HP^{1}, the space of 1dimensional quaternionic subspaces of H^{2}. The "canonical line bundle" over this gives the generator of KS^{4}, or in other words, π_{3}(O(∞)).
Taking n = 7, we can think of S^{8} as OP^{1}, the space of 1dimensional octonionic subspaces of O^{2}. The "canonical line bundle" over this gives the generator of KS^{8}, or in other words, π_{7}(O(∞)).
By Bott periodicity,
π_{7}(O(∞)) = π_{8}(kO) = π_{0}(kO)
so the canonical line bundle over OP^{1} also defines an element of π_{0}(kO). But
π_{0}(kO) = [S^{0},kO] = KS^{0}
and KS^{0} simply records the difference in dimension between the two fibers of a vector bundle over S^{0}, which can be any integer. This is why the octonions are related to dimension.
If for any pointed space we define
K^{n}(X) = K(S^{n} smash X)
we get a cohomology theory called Ktheory, and it turns out that
K^{n+8}(X) = K(X)
which is another say of stating Bott periodicity. Now if * denotes a single point, K(*) is a ring (this is quite common for cohomology theories), and it is generated by elements of degrees 1, 2, 4, and 8. The generator of degree 8 is just the canonical line bundle over OP^{1} and multiplication by this generator gives a map
K^{n}(*) → K^{n+8}(*)
which is an isomorphism of groups  namely, Bott periodicity! In this sense the octonions are responsible for Bott periodicity.
Addendum: The Clifford algebra clock is even better than I described above, because it lets you work out the fancier Clifford algebras C_{p,q}, which are generated by p square roots of 1 and q square roots of 1, which all anticommute with each other. These Clifford algebras are good when you have p dimensions of "space" and q dimensions of "time", and I described the physically important case where q = 1 in "week93". To figure them out, you just work out p  q mod 8, look at what the clock says for that hour, and then take NxN matrices of what you see, with N chosen so that C_{p,q} gets the right dimension, namely 2^{p+q}. So say you're a string theorist and you think there are 9 space dimensions and 1 time dimension. You say: "Okay, 9  1 = 8, so I look and see what's at 8 o'clock. Okay, that's R, the real numbers. But my Clifford algebra C_{9,1} is supposed to have dimension 2^{9 + 1} = 1024 = 32^{2}, so my Clifford algebra must consist of 32x32 matrices with real entries."
By the way, it's not so easy to see that the canonical line bundle over OP^{1} is the generator of KS^{8}  or equivalently, that left multiplication by unit octonions defines a map from S^{7} into SO(8) corresponding to the generator of π_{7}(O(∞)). I claimed it's true above, but when someone asked me why this was true, I realized I couldn't prove it! That made me nervous. So I asked on sci.math.research if it was really true, and I got this reply:
From: Linus Kramer Newsgroups: sci.math.research Subject: π_7(O) and octonions Date: Tue, 09 Nov 1999 12:44:33 +0100 John Baez asked if π_7(O) is generated by the (multiplication by) unit octonions. View this as a question in KOtheory: the claim is that H^8 generates the reduced real Ktheory \tilde KO(S^8) of the 8sphere; the bundle H^8 over S^8 is obtained by the standard glueing process along the equator S^7, using the octonion multiplication. So H^8 is the octonion Hopf bundle. Its Thom space is the projective Cayley plane OP^2. Using this and Hirzebruch's signature theorem, one sees that the Pontrjagin class of H^8 is p_8(H^8)=6x, for a generator x of the 8dimensional integral cohomology of S^8 [a reference for this calulation is my paper 'The topology of smooth projective planes', Arch. Math 63 (1994)]. We have a diagram cplx ch KO(S^8) > K(S^8) > H(S^8) the left arrow is complexification, the second arrow is the Chern character. In dimension 8, these maps form an isomorphism. Now ch(cplx(H^8))=8+x (see the formula in the last paragraph in Husemoller's "Fibre bundles", the chapter on "Bott periodicity and integrality theorems". The constant factor is unimportant, so the answer is yes, π_7(O) is generated by the map S^7> O which sends a unit octonion A to the map l_A:x → Ax in SO(8). Linus KramerMore recently I got an email from Todd Trimble which cites another reference to this fact:
From: Todd Trimble Subject: Hopf bundles To: John Baez Date: Fri, 25 Mar 2005 16:37:11 0500 John, In the book Numbers (GTM 123), there is an article by Hirzebruch where the Bott periodicity result is formulated as saying that the generators of \tilde KO(S^n) in the cases n = 1, 2, 4, 8 are given by [η]  1 where η is the Hopf bundle corresponding to R, C, H, O and 1 is the trivial line bundle over these scalar "fields" (of real dimension 1, 2, 4, 8), and is 0 for n = 3, 5, 6, 7 [p. 294]. Also that the Bott periodicity isomorphism \tilde KO(S^n) > \tilde KO(S^{n+8}) is induced by [η(O)]  1 [p. 295]. I know you are aware of this already (courtesy of the response of Linus Kramers to your sci.math.research query), but I thought you might find a published reference, on the authority of no less than Hirzebruch, handier (should you need it) than referring to a sci.math.research exchange. Unfortunately no proof is given. Hirzebruch says (p. 295), Remark. Our formulation of the Bott periodicity theorem will be found, in essentials, in [reference to Bott's Lectures on K(X), without proofs]. A detailed proof within the framework of Ktheory is given in the textbook [reference to Karoubi's Ktheory]. The reader will have a certain amount of difficulty, however, in extracting the results used here from Karoubi's formulation. Todd
© 1997 John Baez
baez@math.removethis.ucr.andthis.edu
