6 problems

Higher-order Arf invariant conjecture. The maps $\operatorname{Arf}_{j}$ are isomorphisms for all $j$. This is known for $j=1$, where it is given by the classical Arf invariants of…

For a closed oriented or non-orientable manifold product with an Arf invariant normal map, the resulting surgery obstruction modulo the ordinary Arf–Kervaire invariant is called a…

Let $A:S^2\cup S^2\looparrowright B^4$ support a Whitney tower $\mathcal W$, and let $t(\mathcal W)$ denote its associated tree invariant. Let $T$ be the sum of trees in Figure 8 o…

Let $K$ be a knot with non-trivial classical Arf invariant, and let its Bing double be the associated 2-component link. Framed Bing-double conjecture. The Bing double of any such k…

Let $A:S^2\cup S^2\looparrowright B^4$ be a pair of immersed $2$-spheres supporting a twisted Whitney tower $\mathcal W$, and let $t(\mathcal W)$ be its tree invariant. Consider th…

$k$-repeating higher-order Arf conjecture. The map $\mathrm{Arf}^k_j$ is an isomorphism for all $k$ and $j$. This is the $k$-repeating analogue of the higher-order Arf invariant co…