In the previous article we sorted out the closest topological relatives of the sphere in any dimension. We're now ready to look for slightly more distant relatives — and that's what the Kervaire invariant problem is all about.

What if, rather than only allowing transformations that squeeze or stretch a manifold but don't allow cutting or gluing, we admit more drastic alterations? What if we allow ourselves to cut shapes out of manifolds and glue new shapes in along the boundaries of the cut-open original shape? As the image below shows, it then becomes possible to turn a torus into a sphere.

There is actually a clearly defined mathematical way of doing this cutting and gluing in, known as surgery. You could think of surfaces that can be surgically turned into a sphere as sort of distantly related cousins. They are not topologically equivalent to a sphere, but a bit of surgery will get them there.

The Kervaire invariant problem looks for these surgically related cousins of the sphere. It restricts itself to looking for them among manifolds that come with a framing, that is, with some extra information of how the manifold sits within the surrounding higher-dimensional space (you can see the precise definition of a framed manifold on Wikipedia). The question is, given a framed n-dimensional manifold, can it be turned into the n-sphere using surgery —  or not?

Stubborn dimensions

Major headway on the question was achieved in 1969 when the American mathematician  William Browder showed that in "most" dimensions any framed manifold can be turned into a topological sphere using surgery.

The resulting topological sphere would be what is called an exotic sphere, meaning that it possesses a deeper mathematical structure (a differential structure) fundamentally different from that of the true n-sphere. The morphing process that turns an exotic sphere into the true n-sphere involves violent bumps and jerks which this deeper structure can't survive (find out more in this article). It was Milnor who had discovered exotic spheres in the 1950s and who invented the process of surgery to deal with them. But although exotic spheres are, well, exotic, they are still topological spheres.

The dimensions that weren't covered by Browder's work were of a very particular type. The numbers that specified the dimension could be written as $2^k-2$  for integers $k=2,3,4,$ etc, giving the numbers 

$2^2-2=2$

$2^3-2=6$

$2^4-2=14$

$2^5-2=30$

$2^6-2=62$

$2^7-2=126$

$2^8-2=254$

$2^9-2=510$

and so on. This means that for framed 2-manifolds, 6-manifolds, 14-manifolds, 30-manifolds and so on, there was a possibility they'd be so awkward, you couldn't turn them into a  topological sphere even by drilling bits out and gluing bits in. 

Crashing on rocky shores

It was again an invariant that helped with the search for these awkward beasts. In 1960, building on the work of the Turkish mathematician Cahit Arf, Kervaire had defined a number which you could calculate for any given framed manifold. Its value would always be either 0 or 1. If this number evaluated to 0 then the manifold could be surgically converted to a sphere. If it evaluated to 1 then the manifold couldn't be surgically converted to a sphere. The number became known as the Kervaire invariant. It defines two baskets: one for manifolds that are surgically related to the sphere and one for those that are not.

According to Browder's result the Kervaire invariant could only be equal to 1 for manifolds whose dimension was of the type $2^k-2$. This, accordingly, is where mathematicians went looking for those awkward manifolds and they met with some initial success. By the 1980s they had proved that framed manifolds with Kervaire invariant 1 existed in dimensions 2, 6, 14, 30 and 62. 

What obstructs the surgery construction in these cases is their framing, that is, the way they are sitting in the surrounding space. "For a given manifold there could be different framings," says Xu. "With one framing you might be able to convert [the manifold] surgically into a sphere but with another you might not." Even the humble torus comes with a framing that obstructs surgery. Loosely speaking, this framing comes from giving the torus a twist so that it passes through itself. This twist can't be undone by surgery. 

Mathematical nature is often kind in that it confirms patterns mathematicians suspect exist, so people assumed that all the other dimensions on the $2^k-2$ list would follow suit: that all these dimensions harboured framed manifolds of Kervaire invariant 1.

"The dominant view in the community was that these all existed, and most people working in the field had tried to prove this at various points," says Hill. The British mathematician Victor Snaith even published a book about Kervaire invariant 1 manifolds in 2009 which said in the preface, “this might turn out to be a book about things which do not exist.”  The scenario that they might not was named the doomsday hypothesis because it would presumably render a lot of work useless. 

"But people's proofs [that framed manifolds of Kervaire invariant 1 do exist] always fell apart," says Hill. "It was like rocky shores that people kept crashing on." 

Hill, together with Mike Hopkins and Douglas Ravenel, then decided to come at the problem from the side. "We had been working with something called higher real K-theories that Hopkins and Miller had developed," says Hill. These were tools for working in homotopy theory which promised to be powerful, but which people hadn't yet been able to apply in many settings. "We thought, oh wait, I wonder if we can use [these tools], to say something about the Kervaire problem. So we sat down and did the warm-up computations to see what the arc of the story would look like." 

That arc proved to be a huge surprise. Not long after Snaith published his fateful book Hill, Hopkins and Ravenel were able to prove that no manifolds of Kervaire invariant equal to 1 exist in dimensions 254 and higher. "We thought, oh my God, this situation is more complicated than we'd imagined. It was a crazier story than we had thought."

The final frontier: dimension 126

This left one dimension still to be settled: dimension 126. Not long after the breakthrough of Hill and his colleagues, Zhouli Xu was starting his PhD at the University of Chicago under the supervision of Peter May. "[I was chatting in Peter May's office with other new PhD students] when all of a sudden Peter said, 'recently Hill, Hopkins and Ravenel solved the Kervaire invariant problem with one exception: dimension 126. Why don't you solve this? It is going to be your thesis problem.' I thought he was joking." 

But May had been serious — and Xu was right to be daunted. May introduced Xu to Mark Mahowald, a major expert in the field then at Northwestern University who also became Xu's PhD supervisor. "The Kervaire invariant problem sits inside a bigger [area which studies]  stable homotopy groups of spheres," says Xu. "Mahowald had this encyclopaedic knowledge of [the area], not just of the literature: a lot of things are in his mind." And Mahowald confirmed to Xu that dimension 126 was a "life-long problem". 

At the time that Xu started working in the area, in 2011, there was no real sense of how dimension 126 was going to go: whether it harboured framed manifolds of Kervaire invariant 1 or not. The techniques used for dimensions greater than 126 were very different from techniques used for low dimensions. Xu started by familiarising himself thoroughly with dimension 62 — the last dimension before 126 of the required form. What was interesting was that, to prove the result of dimension 62, you did not need to know everything for all the dimensions that went before. "There was a shortcut," says Xu. "You only need to have complete information for about three quarters of the range — up to  dimension about 45 or 47."

To do this Xu needed a good understanding of those stable homotopy groups of spheres — these objects concern ways of relating spheres of different dimension to each other (you can find out more in this article. The problem was that such an understanding was, and still is, one of the greatest challenges in algebraic topology. Ravanel has said that he doesn't expect it to come about in the lifetime of his grandchildren.

At the time Xu started his journey towards the Kervaire invariant problem, rigorous understanding of stable homotopy groups of spheres stretched to dimensions somewhere in the 40s. "Peter May's suggestion was, 'your problem is 126. Three quarters of that is around 90 something. If you can double the range of the knowledge and then look at the shortcut, maybe you can just get to 126.'" 

It's this approach that Xu decided to follow over the next ten years or so, with crucial input from Dan Isakson of Wayne State University, Xu's third PhD supervisor. Collaborators Weinan Lin and Guozhen Wang of Fudan University in Shanghai  entered at different stages of the process, providing sophisticated computer programs for making complex calculations about stable homotopy groups of spheres. 

Alas the shortcut failed, though not for the want of trying. But  Xu, Lin, and Wang were nevertheless able to march up through the dimension, with a last heroic effort spent in dimension 125.  The, finally, in 2024 they were able to a proof: dimension 126 did harbour framed manifolds of Kervaire invariant 1 — which cannot be turned into spheres using surgery. You can hear from Xu himself talking about his struggle for a proof in our podcast.

This finally settled the Kervaire invariant problem for all dimensions: framed manifolds with Kervaire invariant 1 only exist in dimensions 2, 6, 14, 30, 62, and 126. The awkward beasts are therefore quite rare. In all other dimensions the Kervaire invariant of all framed manifolds is 0. 

Getting stuck — but in good company

If the Kervaire invariant problem demonstrates one thing it's that mathematics, these days, is an intensely collaborative subject. Both Hill and Xu talk of frequent meetings, visits, and zoom calls with collaborators, and also the importance of conference and meetings, such as the research programme they are currently taking part in at the INI. "I'm enjoying the programme very much," says Xu. "In fact, I was here before in 2018 or 2019 during a workshop. That was a very successful visit to me and this time too there are many [opportunities] to talk to many people, exchange ideas and explore future directions."

Hill agrees. "I've started new collaborations with people, some of whom I've never met in person before this programme," he says.  "Most of the senior people I knew, but a lot of the earlier career [researchers], I hadn't had the chance to meet before.  It's been great to have a chance to talk to them, to push things I'm thinking about and to expand into other areas." Indeed, the proof of another big problem in the area, the telescope conjecture, was announced in 2023 at another conference organised by the INI. You can find out more here.

Another thing the Kervaire invariant problem shows is that mathematics can be hard and even a tiny bit frustrating. In our interview Xu talked vividly about the many times he got stuck, sometimes for years. But this isn't a reason not to try, it's just a reason to choose the problems you work on carefully.  "You may want to work on something you're genuinely [interested in]. Then you know that, later on when you have the problem worked out, you will be super-excited. You want to work on something you are really into."

About this article

Michael Hill is Ordway Professor at the University of Minnesota.

Zhouli Xu is a Professor in the Department of Mathematics at the University of California, LA.

Marianne Freiberger, Editor of Plus, interviewed Hill and Xu in April and June 2025.

This content forms part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from the collaboration here.

The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.