# Periodicity theorems in (generalized) cohomology theories

It is well-known that topological K-theory is blessed with the Bott periodicity theorem, which specifies an isomorphism between $K^2(X)$ and $K^0(X)$ (where $K^n$ is defined from $K^0$ by taking suspensions). I am wondering if other generalized cohomology theories have their own periodicity theorems, and if there is a general framework for conceptualizing them. I am interested in any substantive answer to this question, but there are two specific avenues for generalization that I am particularly curious about.

The first avenue begins with the Clifford algebra approach to Bott periodicity. This approach relates periodicity in K-theory to a certain natural periodicity present in the theory of complex Clifford algebras, and it generalizes the 8-fold periodicity of real K-theory (corresponding to an 8-fold periodicity in real Clifford algebras). Can one fruitfully generalize the notion of a Clifford algebra, associate to it a generalized cohomology theory, and analogously produce a periodicity theorem?

The second avenue involves Cuntz's proof of Bott periodicity for C*-algebras (which in particular implies topological Bott periodicity by specializing to commutative C*-algebras). Cuntz proves Bott periodicity for any functor from the category of C*-algebras to the category of Abelian groups which is stable (i.e. insensitive to tensoring with the C*-algebra of compact operators on Hilbert space), half exact, and homotopy invariant. The proof uses topological properties of Toeplitz algebras in an essential way. Because of the generality of his approach, I am left wondering if the essential features of his argument can be translated into more general contexts.

Any ideas are welcome!

• Unfortunately, unlike K-theory it happens that most of the time we might know that a cohomology theory is periodic, or we might know a geometric interpretation like in terms of vector bundles, but rarely do we have both. – Tyler Lawson Jul 17 '10 at 18:57

In the spirit of first approach, there is a conjecture for a Clifford-algebra type proof of the 576-fold periodicity of TMF. This is a generalized cohomology theory constructed by piecing together all the elliptic cohomology theories together in a suitable way. I heard about this conjecture from Andre Henriques, who is working on a geometric approach to TMF using conformal nets.

The idea is that the free fermion conformal net (a introduction can be found in this article by Bartels, Douglas and Henriques) is to TMF as the Clifford algebras are to K-theory. For a suitable sense of equivalence, i.e. some generalization of Morita equivalence, $Free(n)$ and $Free(n+576)$ should be equivalent. I believe people are still far from a proof, but the motivation comes from looking at orientations: a manifold is orientable for K-theory if the frame bundle (a principal $SO(n)$-bundle) lifts to a principal $Spin(n)$-bundle. The $Spin(n)$ groups can be defined as a group in the Clifford algebra. For TMF, a manifold is orientable if the frame bundle extends to a principal $String(n)$-bundle, which can be obtained from $Spin(n)$ by killing $\pi_3$, just like $Spin(n)$ is obtained from $SO(n)$ by killing $\pi_1$. There is a way to define $String(n)$ using the free fermion conformal nets.

The only reference I know for these ideas is the following summary.

• This sounds fascinating! I can't wait to find out where the number 576 comes from... – Paul Siegel Jul 18 '10 at 2:57
• 576 is $24^2$. It is supposed to come from points in the moduli stack of elliptic curves with additional automorphisms: those corresponding to j=0 and j=12^3 have Z/6Z and Z/4Z respectively. However, the proof uses spectral sequence arguments at primes 2 and 3: math.mit.edu/conferences/talbot/2007/tmfproc/Chapter17/… – skupers Jul 18 '10 at 12:19

In this video of a lecture given at Atiyah's 80th Birthday Conference, Mike Hopkins gives a description of the solution to the Kervaire Invariant Problem which relies heavily on some periodicity in a cohomology theory created for this problem. He mentions Clifford algebras and Bott Periodicity at one point towards the end of the talk... I don't know a whole lot about this topic so I didn't understand too much, but maybe it will be helpful for you!

• Amusingly enough I was actually there in the audience, and I definitely remember some discussion of generalized periodicity theorems (though I guess I was pretty lost by the end). In any event, it seems I am not the first to wonder about this sort of thing. – Paul Siegel Jul 18 '10 at 3:00

Cohomology theories constructed by the Landweber exact functor theorem are periodic, and there is also a periodization of cohomology theories.

• Is this really the case? all genera lead to periodic theories? – Sean Tilson Aug 8 '10 at 18:38
• @SeanTilson There is e. g. the "tautological" genus with values in the Lazard ring. – მამუკა ჯიბლაძე Aug 4 '17 at 0:20
• @მამუკაჯიბლაძე I am confused by your response. I don't think all genera should lead to periodic theories. – Sean Tilson Aug 12 '17 at 19:55
• @SeanTilson My comment was supposed to give an example of a non-periodic one: namely, complex cobordism. – მამუკა ჯიბლაძე Aug 12 '17 at 22:44
• The assertion that cohomology theories constructed by the Landweber Exact Functor theorem are periodic. This isn't true. To use the LEFT you need a genus... Does this clarify what I meant? – Sean Tilson Aug 15 '17 at 12:28