2 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 $M$ and $N$ be topological $C^k$-manifolds of dimension $m$, let $M$ be immersed in $R^{2m+1}$, and for each $x\in M$ and ambient rotation direction $\mathcal{O}$ suppose that…