The n-Category Café

My personal spy on that conference, Danny Stevenson, mentioned, if I understood him correctly, that Hopkins also said something about the existence of “generalized Clifford algebras” in some of these dimensions. Er, maybe I don’t know in which dimensions, is it these 254, 510, 1022… and then maybe 34560?

Anyway, I am guessing this must be related to (an even higher version of) the higher Clifford algebras discussed here?

One kind of higher Clifford algebra per chromatic filtration/brane dimension?

will turn out to be at least slightly related to theories of 3-branes

Somehow it would have felt better if it had been not 3 but $4 k + 1$ for $k \in \mathbb{N}$. That’s the brane dimensions where special relations to higher Chern-Simons theories, self-dual forms etc. exist, and its the dimensions for which one expects there to be “fundamental” or “NS”-branes: the string, the fundamental 5-brane, maybe the space-filling NS 9-brane.

(Not that I claim to have more then a superficial understanding of the specialty of $4 k +1$ here. )

I don’t get why this would solve the problem of exotic spheres because there is no mention about the existence of exotic spheres in 4 dimensions…

Isn’t the case n = 126 still open?

The slides containing the sketch of the proof are presented here (50MB warning)!. I found it today on wikipedia on Kervaire invariant entry. The file is hosted on Andrew Ranicki’s Webpage.

Personaly, the most mysterious part for me is the second last page, where it asks about a 4 dimensional QFT (what is that?!?), and “generalized Clifford algebras” with peridiocity of 34560.

Here’s part of an email from Mike Hopkins:

First of all, my apologies about the size of the file for my slides. I did the talk in apple’s keynote (and used illustrator for all of the charts and artwork). For some reason keynote always makes HUGE pdf files. I suspect it is because of the “chalkboard” background I chose. I also suspect it was kind of hard to tell what I talked about from the slides alone. The talks were recorded, so the audio (and visual) should be available somewhere.

One thing I wanted to comment on was your discussion of exotic spheres. Way back in the 1930’s when Pontryagin introduced his beautiful generalization of the notion of the “degree” of a map he introduced the basic maneuver of framed surgery and made a tiny mistake. He was trying to compute the cobordism group of stably framed 2-manifolds. He argued that if the genus is zero, the manifold bounds, since the usual 2-sphere bounds, and there is only one stable framing since $\pi_2$ of a Lie group is trivial. He then introduced the notion of framed surgery (he didn’t call it that) to reduce the genus. You have to choose a circle along which to do the surgery, and it has to be chosen so that the framing of the circle you chose actually bounds a framed disk. This defines a map

$$H_1(\Sigma;\mathbb{Z}/2) \to \mathbb{Z}/2$$

and Pontryagin assumed at first that it was linear. In his book on differential topology which appeared much later he noted that it is a quadratic refinement of the intersection pairing, and that the Arf invariant is a cobordism invariant.

Without going through the whole history, let me just say that this exact situation generalizes to dimension $4k+2$. The Arf (Kervaire) invariant is the obstruction to doing framed surgery and reducing your manifold to a sphere. In the language of homotopy theory, there is a map

$$\pi_{4k+2} S^0 \to \mathbb{Z}/2$$

and the kernel consists of (stably) framed manifolds which are framed cobordant to a sphere. This map is zero on the image of the $J$ homomorphism and the Kervaire–Milnor group in dimension $4k+2$ is (if memory serves) the kernel of

$$\pi_{4k+2}/Im J \to \mathbb{Z}/2.$$

We now know that that, except for 5 (or maybe 6) values of $k$, this map is zero. You can think of it as finally completing the reduction of the problem of exotic spheres to homotopy theory, or, as showing that Pontryagin’s original construction only actually fails in 5 or 6 dimensions, and in all other dimensions of the form $(4k+2)$ every framed manifold is framed cobordant to a sphere.

Of course the Kervaire invariant problem had many other forms and played many other roles in algebraic and geometric topology.

One cute formulation is in the front piece of Ioan James’ book on Stiefel Manifolds. He considers the configuration space of ordered pairs of distinct and non-antipodal points $(x,y)$ in $S^n$, and asks if there is a homotopy of the identity map of this space with the map $(x,y) \to (y,x)$. If you look at the case of $S^2$ you’ll get the idea of rotating 180 degrees in the counter-clockwise direction around $(x+y)/|x+y|$. A little thought shows that this would work on $S^n$ if you had an almost complex structure. So this gets you $n=2$ and $n=6$, which are the first two numbers of the form $2^{n-2}$. I’m not sure how one might go further with this, but it’d be nice to do it somehow. I also have to confess to never having worked out the relation of this to the Kervaire invariant, though for some reason I’ve long been under the impression that constructing such a homotopy divides a certain whitehead product by 2, and that that gives you an element of Kervaire invariant 1. Oh, and if you actually look at the book you’ll see that there’s a typo in the statement, and that what he actually states is obviously wrong.

Whew! Didn’t mean to write so much!

Mike

Anyone interested in this new result who will be near Lisbon next week should go to this talk:

• Douglas Ravenel, Recent Developments in Stable Homotopy Theory from the Chromatic Point of View, Room P3.10, Department of Mathematics, Instituto Superior Técnico, Lisbon, May 5-7.
  
  Abstract: In the past three decades, homotopy theorists have found increasingly deep connections between stable homotopy theory and algebraic geometry. This series of talks will outline the necessary background and describe some recent progress that is joint work with Mike Hill and Mike Hopkins. Here is a brief description. The best approach to the stable homotopy groups of spheres is the Adams-Novikov spectral sequence. This is the Adams spectral sequence based on complex cobordism theory ($MU$) or equivalently (after localizing at a prime $p$) Brown-Peterson theory ($BP$). The relevant algebra is controlled by the theory of 1-dimensional formal group laws. This connection was first discovered by Quillen in 1969, then explored more deeply by Morava in the early 1970s. This was followed by the discovery of the chromatic filtration of the stable homotopy category in the 1980s and the work of Hopkins and Miller in the 1990s.
  
  A formal group law over an algebraically closed field in characteristic $p$ is determined up to isomorphism by an invariant called the height, which is a positive integer $n$. A height $n$ formal group law has an automorphism group $S_n$ known as the Morava stabilizer group. It is a pro-$p$-group with interesting arithmetic properties. We now know that it has a canonical action on a certain $E_\infty$ ring spectrum $E_n$ which is difficult to describe explicitly, but very valuable to know. Knowing the cohomology of this action would tell us a lot about stable homotopy, but the problem is prohibitively difficult for $n \gt 2$.
  
  A more tractable problem involves finite subgroups $G$ of $S_n$, which have been classified by Hewett. The most interesting cases are ones with order divisible by $p$, which occur when $n$ is divisible by $p-1$. In these cases we can look at the homotopy fixed point set of $G$ acting on $E_n$. Studying this in depth requires some techniques from equivariant stable homotopy theory, which we will introduce as needed.

Nature magazine has an article on the Kervaire invariant one problem, which is worth reading just for the silly picture. Whoops!

I dimly recall a comment about related work of Akhmetiev but `search’ in this blog doesn’t find it - perhaps due to the transliteration problem.

I followed a link from Doug Ravenel’s page on the Arf-Kervaire invariant to a Scientific American article on the Hill-Hopkins-Ravenel result.

http://www.scientificamerican.com/article.cfm?id=hypersphere-exotica

It looks like John B’s heat of the moment mistake as corrected by Andrew R on this blog has propagated to the pages of Scientific American - that’s my guess anyway. I have tried to correct it there.

The proof was uploaded to arxiv.