Yet more on octonions....

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Keywords: Also available at http://math.ucr.edu/home/baez/week105.html

June 21, 1997

This Week's Finds in Mathematical Physics - Week 105John Baez

There are some spooky facts in mathematics that you'd never guess in amillion years... only when someone carefully works them out do theybecome clear. One of them is called "Bott periodicity".

So, what are the homotopy groups of O(infinity)? Well, they start outlooking like this:

n pi_n(O(infinity))

0 Z/21 Z/22 03 Z4 05 06 07 Z

And then they repeat, modulo 8. Bott periodicity strikes again!

But what do they mean?

But if we keep Bott periodicity in mind, there isanother way to think of it: we can think of it as being about pi_{-1},since 7 = -1 mod 8.

But wait a minute! Since when can we talk about pi_n when n is*negative*?! What's a -1-dimensional sphere, for example?

Well, the idea here is to use a trick. There is a space very relatedto O(infinity), called BO(infinity). As with O(infinity), the homotopygroups of this space repeat modulo 8. Moreover we have:

pi_n(O(infinity)) = pi_{n+1}(BO(infinity))

Combining these facts, we see that the very subtle pi_7 of O(infinity)is nothing but the very unsubtle pi_0 of BO(infinity), which justkeeps track of how many connected components BO(infinity) has.

But what *is* BO(infinity)?

` pi_n(BG) = pi_(n+1)(BG), for n =
0,1,2,...``
`

`Thus, unless you meant otherwise, pi_0(BO(infinity)) = 0, *not*
Z.``
`

`In order properly to understand this Bott periodicity phenomenon
in`
`homotopy groups, one needs to make use of loop spaces: the
loop space`
`LX of a base pointed space X is the space of
all (based) loops in`
`X, ie. the space of all (base point preserving maps) S^1
--> X.``
`

`More generally, there is the n-th order loop space L^n(X),
which`
`is the space of all (base point preserving maps) S^n -->
X. It is`
`quite easy to see that L^(i+j)(X) = L^i(L^j(X)). Of
course, we have`
`L^0(X) = X trivially. There is a fibration PX
--> X, where PX`
`is the space of paths [0,1] --> X originating at the
base point of`
`X, and LX is the homotopy fiber. Hence,
PX is contractible, and``
`

` pi_n(LX) = pi_(n+1)(X), for n = 0,1,2,...``
`

`Note the similarity with the correspoding statement about homotopy`
`groups of a group G and its classifying space
BG. Indeed, the`
`principal G-bundle EG --> BG is homotopy equivalent,
as fibration,`
`to the path fibration L(BG) --> BG, and so G = L(BG),
up to homotopy.``
`

`Now, using L^(i+j)(X) = L^i(L^j(X)), and the above result,
we see`
`immediately that``
`

` pi_n(L^k(X)) = pi_(n+k)(X), for n
= 0,1,2,...``
`

`Now, back to Bott periodicity. In its "coarsest" form, it
says that,`
`up to homotopy,``
`

` L^7(O(infinity)) = Z x BO(infinity)``
`

`or that``
`

` L^8(O(infinity)) = O(infinity)),``
`

`which better displays the 8-fold periodicity. Alternatively,
Bott`
`periodicity says that, up to homotopy,``
`

` L^8(BO(infinity)) = Z x BO(infinity),``
`

`but since L^k(Z x BO(infinity)) = L^k(BO(infinity))
for k > 0,`
`on the nose (ie., they're homeomorphic, not just homotopy`
`equivalent), we have the homotopy equivalence``
`

` L^8(Z x BO(infinity)) = Z x BO(infinity),``
`

`again displaying perfectly the 8-fold periodicity.``
`

`There is a *finer* version of Bott periodicity. It says that,`
`up to homotopy:``
`

`(1) L(Z x BO(inf)) = O(inf)`
`(2) L(O(inf))
= O(inf)/U(inf)`
`(3) L(O(inf)/U(inf)) = U(inf)/Sp(inf)`
`(4) L(U(inf)/Sp(inf)) = Z x BSp(inf)`
`(5) L(Z x BSp(inf)) = Sp(inf)`
`(6) L(Sp(inf))
= Sp(inf)/U(inf)`
`(7) L(Sp(inf)/U(inf)) = U(inf)/O(inf)`
`(8) L(U(inf)/O(inf)) = Z x BO(inf)``
`

`Incidentally, (2-fold) the complex Bott periodicity says`
`that:``
`

`(1c) L(Z x BU(inf)) = U(inf)`
`(2c) L(U(inf))
= Z x BU(inf)``
`

`These were the results actually proved originally by Bott,`
`essentially by a careful examination of certain spaces of`
`smooth loops on classical groups and homogeneous spaces.``
`

`In my paper, "Bott periodicity and the Q-construction,"`
`Contemp. Math. 199(1996), 107-124, I construct actual`
`fibrations with contractible total spaces as follows`
`(the fibers are precise, rather than "fibers up to`
`homotopy):``
`

` base space
fiber`
`(1) Z x BO(inf)
O(inf)`
`(2) O(inf)
O(inf)/U(inf)`
`(3) O(inf)/U(inf) U(inf)/Sp(inf)`
`(4) U(inf)/Sp(inf) Z x BSp(inf)`
`(5) Z x BSp(inf) Sp(inf)`
`(6) Sp(inf)
Sp(inf)/U(inf)`
`(7) Sp(inf)/U(inf) U(inf)/O(inf)`
`(8) U(inf)/O(inf) Z x BO(inf)``
`

`(1c) Z x BU(inf) U(inf)`
`(2c) U(inf)
Z x BU(inf)``
`

`The proof of Bott periodicity via the above fibrations`
`was motivated by considerations from algebraic K-theory.`
`It is also closer, in spirit if not in style or manner,`
`to the broad framework of Bott's original proof.`

That's more or less the end of what I have to say, except for somereferences and some remarks of a more technical nature.

Bott periodicity for O(infinity) was first proved by Raoul Bott in1959. Bott is a wonderful explainer of mathematics and one of themain driving forces behind applications of topology to physics, anda 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: MathematicsRelated to Physics (the 1980s). Birkhauser, Boston, 1994, 2355 pagestotal.

A good paper on the relation between O(infinity) and Clifford algebrasis:

2) M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology(3) 1964, 3-38.

`[snip]`

Let me briefly explain this BO(infinity) business. For anytopological group G you can cook up a space BG whose loop space ishomotopic to G. In other words, the space of (base-point-preserving)maps from S^1 to BG is homotopic to G. It follows that

pi_n(G) = pi_{n+1}(BG).

This space BG is called the classifying space of G because it has aprincipal G-bundle over it, and given *any* decent topological space X(say a CW complex) you can get all principal G-bundles over X (up toisomorphism) by taking a map f: X -> BG and pulling back thisprincipal G-bundle over BG. Moreover, homotopic maps to BG giveisomorphic G-bundles over X this way. Now a principal O(n)-bundle isbasically the same thing as an n-dimensional real vector bundle ---there are obvious ways to go back and forth between these concepts. Aprincipal O(infinity)-bundle is thus very much like a real vectorbundle of *arbitrary* dimension, but where we don't care about addingon arbitrarily many 1-dimensional trivial bundles. If we take thecollection of isomorphism classes of real vector bundles over X anddecree two to be equivalent if they become isomorphic after adding ontrivial bundles, we get something called KX, the "real K-theory of X".It's not hard to see that this is a group. Taking what I've said andworking a bit, it follows that

KX = [X, BO(infinity)]

`[snip]`

If we take X to be S^{n+1}, we see

KS^{n+1} = pi_{n+1}(BO(infinity)) = pi_n(O(infinity))

It follows that we can get all elements of pi_n of O(infinity)from real vector bundles over S^{n+1}.

Taking n = 0, we can think of S^1 as RP^1, the real projective line,i.e. the space of 1-dimensional real subspaces of R^2. This has a"canonical line bundle" over it, that is, a 1-dimensional real vectorbundle which to each point of RP^1 assigns the 1-dimensional subspace ofR^2 that *is* that point. This vector bundle over S^1 gives the generatorof KS^1, or in other words, pi_0(O(infinity)).

Taking n = 1, we can think of S^2 as the "Riemann sphere", or in otherwords CP^1, the space of 1-dimensional complex subspaces of C^2. Thistoo has a "canonical line bundle" over it, which is a 1-dimensionalcomplex vector bundle, or 2-dimensional real vector bundle. Thisbundle over S^2 gives the generator of KS^2, or in other words,pi_1(O(infinity)).

Taking n = 3, we can think of S^4 as HP^1, the space of 1-dimensionalquaternionic subspaces of H^2. The "canonical line bundle" over thisgives the generator of KS^4, or in other words, pi_3(O(infinity)).

`More generally, taking a real vector bundle V over
X and (fiberwise)`
`tensoring it with the canonical quaternionic line bundle
N over HP^1`
`and subtracting V tensored with the trivial quaternitionic
line bundle`
`1_H over HP^1 = S^4 leads to the isomorphism
KO(X) = KSp(S^4(X)).`
`The same trick works for symplectic K-theory KSp; viz., taking
a`
`quaternionic vector bundle W over Y and
tensoring with N* (the`
`conjugate of N) over HP^1 and subracting
1_H over HP^1 = S^4`
`leads to the isomorphism KSp(Y) = KO(S^4(Y)). These
two isomorphisms`
`just described combined to give the K-theoretic proof of Bott`
`periodicity: KO(X) = KO(S^8(X)), KSp(Y) = KSp(S^8(Y))
-- a proof`
`championed by Atiyah in the late 60's. Note that (W
tensor N*) above`
`is only a real bundle -- the tensor product is taken over
H.`

Taking n = 7, we can think of S^8 as OP^1, the space of 1-dimensionaloctonionic subspaces of O^2. The "canonical line bundle" over thisgives the generator of KS^8, or in other words, pi_7(O(infinity)).

`However, the composite isomorphism Z = KO(S^0) = KSp(S^4)
= KO(S^8)`
`above realizes the generator of KO(S^8) differently,
namely as`
`(N tensor N*) - 4_R. So we are talking about (real) 4-plane
bundles`
`here rather than 8-plane bundles. Are you sure your octionionic`
`bundle - 8_R is a generator for KO(S^8) in the following?
...`

By Bott periodicity,

pi_7(O(infinity)) = pi_8(BO(infinity) = pi_0(BO(infinity))

so the canonical line bundle over OP^1 also defines an element ofpi_0(BO(infinity)). But

pi_0(BO(infinity)) = [S^0,BO(infinity)] = KS^0

and KS^0 simply records the *difference in dimension* betweenthe the two fibers of a vector bundle over S^0, which can beany integer. This is why the octonions are related to dimension.

` pi_0(Z x BO(inf)) = [S^0,Z x BO(inf)] = K(S^0).`

If for any pointed space we define

K^n(X) = K(S^n smash X)

we get a cohomology theory called K-theory, and it turns out that

K^{n+8}(X) = K(X)

which is another say of stating Bott periodicity. Now if * denotesa single point, K(*) is a ring (as usual for cohomology theories),

and it is generated by elements of degrees 1, 2, 4, and 8. Thegenerator 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.

`Chuck Giffen`

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