Chinese mathematicians crack 65-year-old Kervaire invariant mystery

Ending a six-decade-old mathematical mystery with the help of computational methods, a trio of Chinese scientists have proven that manifolds of Kervaire invariant one do exist in dimension 126.

The study—authored by Wang Guozhen and Lin Weinan of Fudan University’s Shanghai Center for Mathematical Sciences, and Xu Zhouli of University of California, Los Angeles—is yet to be peer reviewed.

The Kervaire invariant is a mathematical tool for determining whether certain curved shapes, called smooth framed manifolds, can be transformed into spheres using a method known as ‘surgery’. A zero value means they can; while a value of one means they can’t. The Kervaire invariant problem investigates in which dimensions these unusual, non-spherical shapes exist.

Discovery in the 126th dimension

According to the researchers, they have confirmed the existence of smooth framed manifolds with Kervaire invariant one in dimension 126, effectively solving the final unresolved case.

The team concluded that smooth framed manifolds with Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and 126. The problem has puzzled mathematicians for decades. Back in 1963, mathematicians Michel Kervaire and John Milnor proved their existence in dimensions 6 and 14, marking early progress, the South China Morning Post reported.

For decades, mathematicians expected that the pattern would extend to higher dimensions, such as 126 and 254. However, progress stalled at dimension 62. The assumption that Kervaire invariant one manifolds must exist in these higher dimensions influenced the development of propositions about exotic shapes. 

This assumption, however, was eventually challenged, leading to the formulation of the ‘doomsday hypothesis’, which suggested that the expected results might not hold true.

Confirmation of doomsday hypothesis

In 2009, American mathematician Michael Hopkins from Harvard University and his team demonstrated that Kervaire invariant one manifolds exist only in dimensions up to 126 and do not appear in dimensions 254 or higher, confirming the doomsday hypothesis.

Although their proof solved a long-standing problem in algebraic topology, the question of whether Kervaire invariant one manifolds existed in dimension 126 remained unresolved for the next 15 years. Now, the mathematical world has definitive proof that such manifolds do exist in dimension 126, finally closing the chapter on this decades-long mystery.

Hopkins remarked that, prior to their proof being released, mathematicians had considered such a “heroically computational” achievement to be far out of reach. Addressing the dimension 126 problem involved analyzing the stable homotopy groups of spheres, which describe how points on high-dimensional spheres can be mapped or deformed into lower dimensions.

The Adams spectral sequence is a mathematical tool—often visualized as an atlas of dots—that helps researchers navigate the complexities of stable homotopy theory. For dimension 126, it was known that if a specific dot in the 126th column persisted through all stages of this sequence, it would confirm the existence of manifolds in that dimension that can’t be transformed into spheres.

Building on newly developed computational methods by Xu and Wang, Lin designed a program that eliminated 101 of the possible cases. After another year of dedicated work, the team managed to rule out the final four.

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