A team of three has solved a 45-year-old problem in the mathematics of topology.

The Kervaire invariant problem is "one of the major outstanding problems in algebraic and geometric topology" says fellow mathematician Nick Kuhn, at the University of Virginia in Charlottesville.

"Most people thought it would never be solved in their lifetime," adds Mark Hovey, an algebraic topologist at Wesleyan University in Middletown, Connecticut. "Many people have thought they've solved it but have been wrong."

"The solution to this problem seems to indicate new and deep connections between topology on the one hand and algebra and number theory on the other," says mathematician Allen Hatcher of Cornell University in Ithaca, New York. "The exploration of these new connections will enrich the subject for years to come."

Although it looks at face value to be extremely abstruse, the mathematics involved in the solution might be relevant to quantum theory and string theory, not to mention brane theory, which has been invoked to explore some issues in Big Bang cosmology.

## Topological teaser

Mike Hopkins of Harvard University, Douglas Ravenel of the University of Rochester in New York and Mike Hill of the University of Virginia in Charlottesville announced their solution to the problem at a conference in Edinburgh, UK, on 21 April.

“Many people have thought they've solved it but have been wrong.”

Mark Hovey

Wesleyan University

Algebraic topology is a way of describing the properties that objects with the same topology have in common. Topologically equivalent objects are objects that can be converted into each other by deforming but not tearing them: a sphere and an eggshell, for example, or a doughnut and a coffee cup.

Such objects might be curves, surfaces or higher-dimensional entities, collectively called manifolds. Algebraic topology classifies them according to certain invariant quantities related to their geometry. It is "about a relationship between numbers and shapes", says Hopkins.

In 1960 French mathematician Michel Kervaire identified such an invariant for smooth manifolds of dimension n. This Kervaire invariant was in general equal to zero. But it quickly became clear that manifolds with a Kervaire invariant equal to 1 exist in dimensions 2, 6 and 14, and examples for dimensions 30 and 62 were found within a few years.

These numbers share the characteristic that they are all equal to a power of 2 minus 2 (for example, 30 = 2^{5}–2).
In 1969, Princeton mathematician William Browder showed that there can
be no manifolds with non-zero Kervaire invariants outside these special
dimensions.

But what mathematicians really wanted was a general proof of whether the invariant was equal to 1 or 0 for all such values of n.

## Brane science

This is what Hopkins and his colleagues have now found — and they say that the Kervaire invariant is always zero for all n greater than 126. That is in line with what mathematicians intuitively expected. The case of 126 itself is still ambiguous.

The maths used for this solution has been developed by Hopkins, who Hovey calls "clearly the leading algebraic topologist of the day", and others. "The work is based on new conceptual ideas," says Hovey, who adds, "I'm very confident it is correct."

"In some respects the solution of the Kervaire invariant problem is like the proof of Fermat's last theorem in the 1990s," says Hatcher. "The importance lies with the new tools, techniques and insights that were developed to get the solution."

Because the new approach involves looking at topological problems of a manifold from the perspective of a space that has one more dimension, it is analogous to the use of one-dimensional strings as the basis of zero-dimensional (point-like) fundamental particles. Similarly, it has become popular for cosmologists to study the behaviour of space-time from the perspective of higher-dimensional 'branes' that interact with one another. This is why studying the Kervaire invariant problem might offer useful mathematical techniques to fundamental physics.

For a donut and a coffee cup to be topologically equivalent, said cup must have a closed circuit handle enclosing a void, as does the donut. The example pictured is invalid. Substitute something by Wedgewood. Are a hat band and a Moebius strip topologically equivalent? (Given the parity of the former's generating functions, is its apparent chirality only an artifact of construction?)

Yes, Uncle Al, I was certainly a bit confused by the statement 'a donut and a coffee cup are equivalent' after seeing the misleading picture given. Perhaps if the donut biter had bitten all the way through the ring it would have been more accurate!

Not just "a conference in Edinburgh" but the Atiyah80 conference celebrating the 80th birthday of Sir Michael Atiyah, http://www.maths.ed.ac.uk/~aar/atiyah80.htm

The picture should show an ordinary mug not a disposable one.

Apologies for the misleading picture, which has now been replaced. Thanks!

A nice animation of a donut turning into a mug (and back) can be found in the Topology wikipedia article, or directly here: http://upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gif

You say "That is in line with what mathematicians intuitively expected," but that is not correct. In the 70s many mathematicians tried to prove the opposite statement, that manifolds with Kervaire invariant one exist in all dimensions allowed by Browder's theorem. I do not know of anyone, myself included, who expected the answer that we found.