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 105
John Baez

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".

[snip]
So, what are the homotopy groups of O(infinity)?  Well, they start out
looking like this:

n     pi_n(O(infinity))

0         Z/2
1         Z/2
2          0
3          Z
4          0
5          0
6          0
7          Z

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

But what do they mean?

[snip]
But if we keep Bott periodicity in mind, there is
another 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 related
to O(infinity), called BO(infinity).  As with O(infinity), the homotopy
groups of this space repeat modulo 8.  Moreover we have:

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

-- for  n = 0,1,2,...
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 just
keeps track of how many connected components BO(infinity) has.
Unless you are using nonstandard terminology, BO(infinity)  has only*one* connected component.  There's nothing subtle about  pi_0  of
BO(infinity) at all -- it's zero.  The space you need is really
Z x BO(infinity).
But what *is* BO(infinity)?
Ah, that's the rub!  For any (discrete or topological) group  G,  BGdenotes the "classifying space" of  G.  It is a *connected* space with
base point which has over it a principal  G-bundle  EG --> BG  whose
total space  EG  is contractible.  Hence, from the homotopy exact
sequence of a fibration, one deduces that

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 some
references and some remarks of a more technical nature.

Bott periodicity for O(infinity) 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(infinity) and Clifford algebras
is:

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

This article reflects subsequent reflection by Bott and his coauthorson Bott's original periodicity theorem, and it contains the 8-fold
periodicity for real Clifford algebras (and Clifford modules, which
you didn't mention, although they are the key to giving an alternate
proof of Bott periodicity).

[snip]

Let me briefly explain this BO(infinity) business.  For any
topological group G you can cook up a space BG whose loop space is
homotopic 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).

for  n = 0,1,2,...
This space BG is called the classifying space of G because it has a
principal 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 to
isomorphism) by taking a map f: X -> BG and pulling back this
principal G-bundle over BG.  Moreover, homotopic maps to BG give
isomorphic G-bundles over X this way.  Now a principal O(n)-bundle is
basically the same thing as an n-dimensional real vector bundle ---
there are obvious ways to go back and forth between these concepts.  A
principal O(infinity)-bundle is thus very much like a real vector
bundle of *arbitrary* dimension, but where we don't care about adding
on arbitrarily many 1-dimensional 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 K-theory 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(infinity)]

-- provided  X  is connected *and* you mean what is known as *reduced*K-theory.  BTW,  KX  usually refers to the complex bundle version, and
what you are calling  KX  is usually denoted  KO(X).

[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 vector
bundle which to each point of RP^1 assigns the 1-dimensional subspace of
R^2 that *is* that point.   This vector bundle over S^1 gives the generator
of KS^1, or in other words, pi_0(O(infinity)).

-- actually, as an element of  KS^1,  you mean the formal differenceL - 1_R  where  L  is the canonical line bundle and  k_R  denotes the
trivial real  k-plane bundle.
Taking n = 1, we can think of S^2 as the "Riemann sphere", or in other
words CP^1, the space of 1-dimensional complex subspaces of C^2.  This
too has a "canonical line bundle" over it, which is a 1-dimensional
complex vector bundle, or 2-dimensional real vector bundle.  This
bundle over S^2 gives the generator of KS^2, or in other words,
pi_1(O(infinity)).
-- once again, you mean something like  M - 1_C, where  M  is the(underlying real 2-plane bundle of the) canonical complex line
bundle over  CP^1.
Taking n = 3, we can think of S^4 as HP^1, the space of 1-dimensional
quaternionic subspaces of H^2.  The "canonical line bundle" over this
gives the generator of KS^4, or in other words, pi_3(O(infinity)).
-- similar coment,  N - 1_H.

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-dimensional
octonionic subspaces of O^2.  The "canonical line bundle" over this
gives the generator of KS^8, or in other words, pi_7(O(infinity)).
DANGER! -- O^2  isn't exactly an  "O-vector space," because  O  isn'tassociative.  There is, however, an octonionic projective line  OP^1
and even an octonionic projective plane  OP^2 -- but *no* octonionic
projective space  OP^n  for  n > 2.  You are asserting that an
"octonionic canonical line bundle" over  OP^1  provides (presumably
as its underlying real bundle -- minus the trivial real 8-plane bundle
8_R, to make the virtual dimension zero as required by reduced
K-theory?) a generator for  KO(S^8).

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))

the last group should of course be  pi_0(Z x BO(inf)) = Z.
so the canonical line bundle over OP^1 also defines an element of
pi_0(BO(infinity)).  But

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

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

You've wandered astray at this point, thanks to not keepingcareful track of what (reduced)  K-theory is.  Indeed, it is
here's where you see that  KX = [X,BO(inf)]  only works for
connected  X,  but  KX = [X, Z x BO(inf)]  is required for
non-connected  X.  Thus, the proper statement is

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 * denotes
a single point, K(*) is a ring (as usual for cohomology theories),

-- of course, *here*,  K(*)  refers to *unreduced* K-theory of apoint, which is isomorphic to *reducedI  K(S^0).  You're not being
very careful here.
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.

-- again, heed the WARNING above.  You're argument as it standsmay seem somewhat specious.  One must be *very* careful on this
point!

Chuck Giffen
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