18 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…

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…

Higher obstruction equivalence conjecture. Such data determines a point in

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…

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…

Let $\mathcal{T}^{\mathrm{tw}}_n$ be the group of order-$n$ twisted Whitney-tower intersection trees, let $\mathsf{W}^{\mathrm{tw}}_n$ be the corresponding graded quotient of links…

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 $A$ be an immersion of $2$-spheres into a $4$-manifold, let $\mathrm{INT}_n(A)$ denote the relations needed to make the obstruction $4\lambda_n(\mathcal{W})4$ independent of th…

A link is height $3.5$ Whitney tower concordant to the Hopf link if it is related to the Hopf link by a height $3.5$ Whitney tower concordance. Friedl and Powell constructed a Cass…

Let $k$ be a positive integer, and consider the horizontal Whitney-tower sequence in which ${\sf K}^{\mu}_{4k-3}$ is the middle group. Let ${\sf L}'_{2k}$ denote the relevant degre…

Let $L$ be a link in $S^3$, and let the obstructions of Theorem be those defined in the paper for detecting whether $L$ is concordant to the Hopf link. Whitney tower/Grope obstruct…

For each $k$, let $\mathcal{T}^{\infty}_{4k-2}$ be the twisted tree group and let $\text{\sf W}^{\infty}_{4k-2}$ be the corresponding twisted graded group of link concordance class…

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…