10 problems

Let $A_1,\dots,A_m$ be the original maps, let $\mathcal{W}$ be a Whitney tower on them, and let $\tau_n(\mathcal{W})$ be its intersection tree, regarded as an element of the quotie…

Let $X$ be a simply connected $4$-manifold, and let $A=A_1,A_2,A_3,A_4$ support an order-$2$ non-repeating Whitney tower $\mathcal W$. Let $\lambda_2(A)$ be the order-$2$ intersect…

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…

Consider the sum of the trees depicted in Figure 8 of the source, associated to 2-component string links, with coefficients in $\mathbb Z/2\mathbb Z$. Framed Arf finite-type conjec…

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…

$k$-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…

Let $X$ be a $4$-manifold and let $A:\amalg^4 S^2\to X$ be a quadruple of immersed $2$-spheres. Suppose that $A$ admits an order $2$ non-repeating Whitney tower $\mathcal{W}$, and…

Let $A:\Sigma\to X$ be a collection of immersed $2$-spheres in a $4$-manifold, and suppose that $\lambda_n(A):=\lambda_n(\mathcal{W})$ is defined using an order $n$ non-repeating W…

Let $\mathcal{T}_n$ be the tree group of order $n$, let $\text{\sf W}_n$ be the corresponding graded group of link concordance classes, and set…