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K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of the Bott periodicity theorem for topological complex K-theory that I've found in the literature:

  • Bott's original proof used Morse theory, which reappeared in Milnor's book Morse Theory in a much less condensed form.
  • Pressley and Segal managed to produce the homotopy inverse of the usual Bott map as a corollary in their book Loop Groups.
  • Behrens recently produced a novel proof based on Aguilar and Prieto, which shows that various relevant maps are quasifibrations, therefore inducing the right maps on homotopy and resulting in Bott periodicity.
  • Snaith showed that $BU$ is homotopy equivalent to $CP^\infty$ once you adjoin an invertible element. (He and Gepner also recently showed that this works in the motivic setting too, though this other proof relies on the reader having already seen Bott periodicity for motivic complex K-theory.)
  • Atiyah, Bott, and Shapiro in their seminal paper titled Clifford Modules produced an algebraic proof of the periodicity theorem. EDIT: Whoops x2! They proved the periodicity of the Grothendieck group of Clifford modules, as cdouglas points out, then used topological periodicity to connect back up with $BU$. Wood later gave a more general discussion of this in Banach algebras and Bott periodicity.
  • Atiyah and Bott produced a proof using elementary methods, which boils down to thinking hard about matrix arithmetic and clutching functions. Variations on this have been reproduced in lots of books, e.g., Switzer's Algebraic Topology: Homotopy and Homology.
  • A proof of the periodicity theorem also appears in Atiyah's book K-Theory, which makes use of some basic facts about Fredholm operators. A differently flavored proof that also rests on Fredholm operators appears in Atiyah's paper Algebraic topology and operations on Hilbert space.
  • Atiyah wrote a paper titled Bott Periodicity and the Index of Elliptic Operators that uses his index theorem; this one is particularly nice, since it additionally specifies a fairly minimal set of conditions for a map to be the inverse of the Bott map.
  • Seminaire Cartan in the winter of '59-'60 produced a proof of the periodicity theorem using "only standard techniques from homotopy theory," which I haven't looked into too deeply, but I know it's around.

Now, for my question: the proofs of the periodicity theorem that make use of index theory are in some vague sense appealing to the existence of various Thom isomorphisms. It seems reasonable to expect that one could produce a proof of Bott periodicity that explicitly makes use of the facts that:

  1. The Thom space of the tautological line bundle over $CP^n$ is homeomorphic to $CP^{n+1}$.
  2. Taking a colimit, the Thom space of the tautological line bundle over $CP^\infty$ (call it $L$) is homeomorphic to $CP^\infty$.
  3. The Thom space of the difference bundle $(L - 1)$ over $CP^\infty$ is, stably, $\Sigma^{-2} CP^\infty$. This seems to me like a route to producing a representative of the Bott map. Ideally, it would even have good enough properties to produce another proof of the periodicity theorem.

But I can't find anything about this in the literature. Any ideas on how to squeeze a proof out of this -- or, better yet, any ideas about where I can find someone who's already done the squeezing?

Hope this isn't less of this is nonsense!

-- edit --

Given the positive response but lack of answers, I thought I ought to broaden the question a bit to start discussion. What I was originally looking for was a moral proof of the periodicity theorem -- something short that I could show to someone with a little knowledge of stable homotopy as why we should expect the whole thing to be true. The proofs labeled as elementary contained too much matrix algebra to fit into parlor talk, while the proofs with Fredholm operators didn't seem -- uh -- homotopy-y enough. While this business with Thom spaces over $CP^\infty$ seemed like a good place to look, I knew it probably wasn't the only place. In light of Lawson's response, now I'm sure it isn't the only place!

So: does anyone have a good Bott periodicity punchline, aimed at a homotopy theorist?

(Note: I'll probably reserve the accepted answer flag for something addressing the original question.)

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Don't have an answer for you, but I'd also like to advertise Bruno Harris' nice little proof of complex Bott periodicity in "Bott periodicity via simplicial spaces." –  Tyler Lawson Dec 13 '09 at 21:07
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+1 for writing such great background to your question! –  Scott Morrison Dec 13 '09 at 22:15
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@ Greg: I don't think it makes sense in general, but if you're willing to work with stablized Thom spaces it can be reasonably defined when subtracting off copies of the trivial bundle. Since in the traditional setting we have $T(V + n) = \Sigma^{2n} TV$ for a vector bundle V, we ought to write $T(V - n)$ for $\Sigma^{-2n} TV$, just to make additivity work. This is more carefully justified in and around prop. 5.16 in Rudyak's book. @ YBL: No, I haven't, but I will now. –  Eric Peterson Dec 16 '09 at 17:46
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A small point: Atiyah-Bott-Shapiro did not prove Bott periodicity in their paper on Clifford Modules, but rather used Bott periodicity to show that the Grothendieck group of Clifford modules is the K-theory group. In fact, they specifically mention that it would be nice to have a proof of Bott periodicity that followed from the algebraic periodicity of Clifford modules. –  cdouglas Dec 18 '09 at 1:12
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Someone should at least mention Dyer and Lashof's proof: Dyer, E.; Lashof, R. A topological proof of the Bott periodicity theorems. Ann. Mat. Pura Appl. (4) 54 1961 231--254. –  Robert Bruner Sep 3 '10 at 0:39

7 Answers 7

up vote 29 down vote accepted

Here is my attempt to address Eric's actual question. Given a real $n$-dimensional vector bundle $E$ on a space $X$, there is an associated Thom space that can be understood as a twisted $n$-fold suspension $\Sigma^E X$. (If $E$ is trivial then it is a usual $n$-fold suspension $\Sigma^n X$.) In particular, if $E=L$ is a complex line bundle, it is a twisted double suspension. In particular, if $X = \mathbb{C}P^\infty$, the twisted double suspension of the tautological line bundle $L$ satisfies the equation $$\Sigma^L\mathbb{C}P^\infty = \mathbb{C}P^\infty.$$ As I understand it, Eric wants to know whether this periodicity can be interpreted as a Bott map, maybe after some modification, and then used to prove Bott periodicity. What I am saying matches Eric's steps 1 and 2. Step 3 is a modification to make the map look more like Bott periodicity.

I think that the answer is a qualified no. On the face of it, Eric's map does not carry the same information as the Bott map. Bott periodicity is a theorem about unitary groups and their classifying spaces. What Eric has in mind, as I understand now, is a result of Snaith that constructs a spectrum equivalent to the Bott spectrum for complex K-theory by modifying $\mathbb{C}P^\infty$. Snaith's model has been called "Snaith periodicity", but the existing arguments that it is the same are a use and not a proof of Bott periodicity. (In that sense, Snaith's model is stone soup, although that metaphor is not really fair to his good paper.)

For context, here is a quick definition of Bott's beautiful map as Bott constructed it in the Annals is beautiful. In my opinion, it doesn't particularly need simplification. The map generalizes the suspension relation $\Sigma S^n = S^{n+1}$. You do not need Morse theory to define it; Morse theory is used only to prove homotopy equivalence. Bott's definition: Suppose that $M$ is a compact symmetric space with two points $p$ and $q$ that are connected by many shortest geodesics in the same homotopy class. Then the set of these geodesics is another symmetric space $M'$, and there is an obvious map $\Sigma M' \to M$ that takes the suspension points to $p$ and $q$ and interpolates linearly. For example, if $p$ and $q$ are antipodal points of a round sphere $M = S^{n+1}$, the map is $\Sigma(S^n) \to S^{n+1}$. For complex K-theory, Bott uses $M = U(2n)$, $p = q = I_{2n}$, and geodesics equivalent to the geodesic $\gamma(t) = I_n \oplus \exp(i t) I_n$, with $0 \le t \le 2\pi$. The map is then $$\Sigma (U(2n)/U(n)^2) \to U(2n).$$ The argument of the left side approximates the classifying space $BU(n)$. Bott show that this map is a homotopy equivalence up to degree $2n$. Of course, you get the nicest result if you take $n \to \infty$. Also, to complete Bott periodicity, you need a clutching function map $\Sigma(U(n)) \to BU(n)$, which exists for any compact group. (If you apply the general setup to $M = G$ for a simply connected, compact Lie group, Bott's structure theorem shows that $\pi_2(G)$ is trivial; c.f. this related MO question.)

At first glance, Eric's twisted suspension is very different. It exists for $\mathbb{C}P^\infty = BU(1)$, and of course $\mathbb{C}P^\infty$ is a $K(\mathbb{Z},2)$ space with a totally different homotopy structure from $BU(\infty)$. Moreover, twisted suspensions aren't adjoint to ordinary delooping. Instead, the space of maps $\Sigma^L X \to Y$ is adjoint to sections of a bundle over $X$ with fiber $\mathcal{L}^2 Y$. The homotopy structure of the twisted suspension depends on the choice of $L$. For instance, if $X = S^2$ and $L$ is trivial, then $\Sigma^L S^2 = S^4$ is the usual suspension. But if $L$ has Chern number 1, then $\Sigma^L S^2 = \mathbb{C}P^2$, as Eric computed.

However, in Snaith's paper all of that gets washed away by taking infinitely many suspensions to form $\Sigma_+^{\infty}\mathbb{C}P^\infty$, and then as Eric says adjoining an inverse to a Bott element $\beta$. (I think that the "+" subscript just denotes choosing a base point.) You can see what is coming just from the rational homotopy groups of $\Sigma^\infty \mathbb{C}P^\infty$. Serre proved that the stable homotopy of a CW complex $K$ are just the rational homology $H_*(K,\mathbb{Q})$. (This is related to the theorem that stable homotopy groups of spheres are finite.) Moreover, in stable, rational homotopy, twisted and untwisted suspension become the same. So Snaith's model is built from the fact that the homology of $\mathbb{C}P^\infty$ equals the homotopy of $BU(\infty)$. Moreover, there is an important determinant map $$\det:BU(\infty) \to BU(1) = \mathbb{C}P^\infty$$ that takes the direct sum operation for bundles to tensor multiplication of line bundles. Snaith makes a moral inverse to this map (and not just in rational homology).

Still, searching for a purely homotopy-theoretic proof of Bott periodicity is like searching for a purely algebraic proof of the fundamental theorem of algebra. The fundamental theorem of algebra is not a purely algebraic statement! It is an analytic theorem with an algebraic conclusion, since the complex numbers are defined analytically. The best you can do is a mostly algebraic proof, using some minimal analytic information such as that $\mathbb{R}$ is real-closed using the intermediate value theorem. Likewise, Bott periodicity is not a purely homotopy-theoretic theorem; it is a Lie-theoretic theorem with a homotopy-theoretic conclusion. Likewise, the best you can do is a mostly homotopy-theoretic proof that carefully uses as little Lie theory as possible. The proof by Bruno Harris fits this description. Maybe you could also prove it by reversing Snaith's theorem, but you would still need to explain what facts you use about the unitary groups.

(The answer is significantly revised now that I know more about Snaith's result.)

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This is very much along the lines of what I was thinking. However, Snaith (and now Gepner, and probably other people) have a more cheerful view of $\Sigma^\infty CP^\infty$ -- namely, they show that $\Sigma^\infty_+ CP^\infty[\beta^{-1}]$ is homotopy equivalent to $BU(\infty)$, where $\beta$ is the element of $\pi_2 \Sigma^\infty_+ CP^\infty$ classifying the reduced tautological bundle $(L - 1)$ over $CP^1 = S^1$. That is, $\Sigma^\infty_+ CP^\infty$ is a lot like $BU(\infty)$, up to invertibility of the Bott element. That said, stone soup is probably still applicable. –  Eric Peterson Dec 17 '09 at 6:10
    
Well maybe my last paragraph is just my mathematical ignorance. In this Snaith result, I wouldn't know what is being proved from what. That is, I wouldn't know whether it could be a proof of Bott periodicity or a theorem that uses Bott periodicity to prove something else. –  Greg Kuperberg Dec 17 '09 at 6:35
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The map $\Sigma^\infty_+ CP^\infty[\beta^{-1}] \to BU(\infty)$ is producible without periodicity. To show that it is an equivalence, Gepner (with a more conceptual proof) does use periodicity, but I don't recall if Snaith (with a more computational proof) uses periodicity or not. I suspect he does; I'll look when I have time tomorrow afternoon. Certainly knowing that this map is a homotopy equivalence implies the periodicity of $BU(\infty)$, since it specifically names an invertible degree 2 element in its homotopy. Also, I meant to say thanks in the previous comment! This is all helpful. –  Eric Peterson Dec 17 '09 at 7:20
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You are more than welcome! Lest people think that I already knew all of this stuff, I certainly didn't. These reviews are a great way for me to learn the material. (I did just last month attend a talk by Mike Freedman that discussed Bott's proof. At the time, it went in one ear and out the other.) –  Greg Kuperberg Dec 17 '09 at 7:28
    
By the way, the key issue remaining is addressed in Snaith's paper, "Localized stable homotopy of some classifying spaces". journals.cambridge.org/action/… –  Greg Kuperberg Dec 17 '09 at 7:37

By studying natural filtrations of loop groups, viewed as affine Grassmannians, S. A. Mitchell, in "The Bott Filtration of a Loop Group", describes an elegant and in its way elementary proof of Bott periodicity. Homotopy theorists will appreciate that it comes down to the inclusion of a cellular skeleton being highly connected, while combinatorist and Lie theorists delight when Mitchell reads off Bott periodicity from the Dynkin diagrams.

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I haven't seen this one before. Thanks! –  Eric Peterson Dec 18 '09 at 2:50

I want to add a thought about this question.

The proof you're suggesting does seem very much along the lines of trying to turn Snaith's method, which shows K-theory is recoverable in some way, and turn it into a proof where you start by knowing nothing about K-theory and then show that Snaith's construction (a) tells you about vector bundles, and (b) tells you the resulting K-groups of a point.

This seems very difficult.

In fact, there is famous generalization of the Goerss-Hopkins-Miller theorem due to Lurie that's based on something like Snaith's theorem - and one key that it requires is that we already know the existence of Lubin-Tate cohomology theories so that we have something to compare our resulting object-with-a-formally-inverted-preorientation to at the end of the day.

So: if you ever figure out how to make this proof go in the K-theory case, I'd like to hear from you.

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Well, I'm no Lurie. Now that I have a firmer idea of what's involved, I'll set this aside for now, possibly returning much later in life to remind myself of how difficult it is. I would at least like to check my half-baked assertion to Greg about the relation of the induced map from the Thom space to the map used to build Snaith's localization; that doesn't seem like it should be so difficult. // Also, thanks again for the Harris reference. That paper really is nice! –  Eric Peterson Dec 18 '09 at 16:41

There is also Atiyah and Singer's proof in "Index theory for skew-adjoint fredholm operators" Inst. Hautes Études Sci. Publ. Math. No. 37 1969 5–26. 57.50

This proof uses Kuiper's theorem on the contractibility of the Unitary group as its silver bullet.

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Ah, Kuiper's theorem. My PhD supervisor said that it's mysterious as to where the proof came from... –  David Roberts Mar 8 '11 at 3:27

My favourite proof of complex Bott periodicity is in Suslin's paper "The Beĭlinson spectral sequence for the $K$-theory of the field of real numbers" translated in J. Soviet Math. 63 (1993), no. 1, 57–58. It uses Segal's $\Gamma$-space machinery to produce infinite loop spaces. Once you accept this machinery (something every homotopy theorist should know about anyway), Bott-periodicity falls out; there is nothing to prove!

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Is this any different from Bruno Harris's proof, which uses the group completion theorem? Does Suslin get real Bott periodicity this way? That would be pretty amazing (especially in two pages, whose main purpose seems to be proving something else!). Harris only does the complex case. –  Dan Ramras Jul 2 '11 at 4:57
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I finally got a copy of Suslin's paper. As far as I can tell, his argument is the same as Bruno Harris's proof (mentioned elsewhere on this page); Suslin just says it much more quickly and doesn't explicitly mention the use of the group completion theorem, which is needed to identify $\Omega B(\coprod BU(n))$ with $\mathbb{Z} \times$ colim $BU(n)$. –  Dan Ramras Jul 15 '11 at 18:00

You might also like Giffen's categorical proof using the Q construction:

Giffen, Charles H. Bott periodicity and the Q-construction. Algebraic K-theory (Pozna´n, 1995), 107–124, Contemp. Math., 199, Amer. Math. Soc., Providence, RI, 1996.

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I like very much the idea of Dusa McDuff to strip the Atiyah-Singer approach of any analysis. In a sense one might say that it is an elaboration on $e^{2\pi i}=1$ :)

She considers restriction of the exponential map $\textrm{Exp}:\mathbb H\to\mathbb U$ from Hermitian operators to unitary operators to the subspace $\mathbb H_{[0,1]}\subset\mathbb H$ of weakly contracting operators (that is, operators with eigenvalues in [0,1]). Fibre over $U\in\mathbb U$ turns out to be the set of those $H\in\mathbb H_{[0,1]}$ for which the space $\textrm{Fix}(H)$ of fixed points of $H$ is a subspace of $\textrm{Fix}(U)$. (Very roughly, "$e^{2\pi iH}$ is that unitary thing which is like 1 wherever $H$ is like 1".)

Moreover any subspace of $\textrm{Fix}(U)$ can be realized as $\textrm{Fix}(H)$ for a unique weakly contracting $H$ with $\textrm{Exp}(H)=U$. This establishes a homeomorphism between the fibre over $U$ and the set of subspaces of $\textrm{Fix}(U)$, i. e. the Grassmanian of $\textrm{Fix}(U)$.

After some massaging one obtains a (quasi)fibration with base the unitary group, fibre the Grassmanian, and contractible total space.

This is in the last section of McDuff's 1977 paper "Configuration spaces"; a detailed proof with the extension to real Bott periodicity by Behrens is here, although there are some flaws later corrected in an addendum. That proof is itself a cleanup of a 1999 version by Aguilar and Prieto.

PS I later realized that some additional care is needed for the eigenvalue 0 (of $H$)...

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