Worked solution: Hill-Hopkins-Ravenel proof via equivariant stable homotopy theory (2009)
Before tackling the general question, mathematicians simply rolled up their sleeves and built examples by hand for the first few cases. The classical Hopf maps immediately settle ; for (dimensions and ) it took decades of ever more intricate explicit constructions.
These hand-built examples served two purposes: they proved the phenomenon is not vacuous (Kervaire invariant genuinely occurs), and — crucially — they told the later general theory exactly where it was NOT allowed to prove non-existence, since any general argument had better not accidentally contradict these five known examples.
For , corresponding to dimensions , explicit framed manifolds of Kervaire invariant were constructed by hand (dimensions classically, via the Hopf invariant one elements; and completed by 1984 through work of Michael Barratt, John Jones, and Mark Mahowald using elaborate Toda bracket and secondary cohomology operation calculations).
No such construction was known for any , leaving the problem wide open for higher dimensions — and the difficulty of these last two cases (each representing years of specialised computation) was already a strong hint that a fundamentally different, more structural approach would be needed to say anything about in general.
- Toda bracket
- A higher, secondary operation in stable homotopy theory, defined on triples of maps whose pairwise composites are already null-homotopic; it detects finer relations invisible to ordinary composition and was a key tool in the hand-built constructions for .