3 problems

Kervaire's classification conjecture. The $5$-th Betti number $B_5(M)$ and Kervaire invariant $\Phi(M)\in \mathbb{Z}_2$ are a complete set of invariants of the homotopy type of $M$…

Let $n=126$. The Kervaire problem asks whether there exists a framed manifold of dimension $2^{n+1}-2$ with Kervaire invariant one. Snaith's conjecture. The Kervaire problem is neg…

Let $b=b(k)$ be the number of ones in the binary expansion of $k$, and assume $n+2\geq 2^{b(k)}$. Let $s=n-2b(k)+k+3$. Consider a skew-framed immersion … of a manifold $M^{b(k)-1}$…