161 problems

A knot $K$ is topologically slice if it bounds a topological slice disk $\Delta$ in $B^4$. It is topologically homotopy ribbon if it bounds such a disk for which the inclusion-indu…

Let $\Delta^4$ be a Poincare ball, and let $\Delta^5$ be a contractible 5-manifold. A handle decomposition has only 0-, 1- and 2-handles when all its handles have these indices. 0-…

A Gluck ball is the Poincare 4-ball associated with a Gluck-twisted $S^4$. Gluck ball 5-ball conjecture. If $\Delta^4$ is a Gluck ball, then … This is described as a special case o…

A Poincare 4-ball is a contractible 4-manifold whose boundary is $S^3$. A Schoenflies 4-ball is a Poincare 4-ball that embeds in $S^4$. Poincare ball 5-ball conjecture. If…

Let $\Delta$ be a Schoenflies ball, and let $-\Delta$ denote its complementary Schoenflies ball in $S^4$. Write $\bar\Delta$ for $\Delta$ with the opposite orientation. Negative-eq…

Expansion conjecture. The space $Y$ smoothly embeds in $S^4$ if and only if it is obtained by a possibly empty sequence of expansions from some $Y'$ of the displayed form, with

Let $c$ be a natural number, and consider irreducible simply connected $4$-manifolds of complexity $c$. Finiteness conjecture. For every natural number $c$ there are finitely many…

Abelian rank conjecture. The group $G$ must be abelian of rank at most $2$.

Let $S$ be a rational algebraic surface, let $C$ be the first Chern class of a differentiable complex vector bundle on $S$, and let $e$ be the corresponding Donaldson-invariant gra…

Intersection-lattice conjecture. The conclusion of Corollary still holds, under the same assumptions, if $X$ is allowed to range over the set of all symplectic fillings of $Y$.

A Moishezon 4-manifold is the smooth 4-manifold constructed from a semi-algebraic cuspidal curve in $\mathbb P^2$ and an epimorphism to a symmetric group, with a smooth map to…

Extension conjecture. The intersection form on $\pi_2M$ is extended from the integers.

Rational blowup and torus-surgery conjecture. $X$ and $Y$ are related via a sequence of surgeries on tori of square $0$, rational blowups, and rational blowdowns.

Torus-surgery conjecture. $X$ and $Y$ are related via a sequence of surgeries on tori of square $0$.

Free-fundamental-group conjecture. Then $X^4$ is (simple) homotopy equivalent to an ANR homology $4$--manifold.

Quinn's conjecture. If $s(X^4)=0$, then $X^4$ is (simple) homotopy equivalent to an ANR homology $4$--manifold.

Weak generalized Property R conjecture. Then

Let $M$ be a topological $4$-manifold, let ${\vee}^3 S^1$ denote the wedge of three circles, and let ${\mathcal S}^0(Wh(Bor))$ denote the zero-framed surgery on the Whitehead doubl…

Weak BMY Conjecture. For each finitely presented group $G$,

Let ${\rm Wh}({\rm Bor})$ denote the untwisted Whitehead double of the Borromean Rings. A link is freely topologically slice if its components bound disjoint locally flat disks in…

Let $G$ be a group, and let $M$ be a symplectic 4-manifold with $\pi_1(M)\cong G$ that minimizes $\chi:\mathfrak{M}(G)\rightarrow {\mathbb Z}$. Let $Q_M$ denote the intersection fo…

Let $M$ and $N$ be closed, orientable, topological $4$-manifolds that are homotopy equivalent, have torsion-free fundamental group, and have the same Kirby–Siebenmann invariant. St…

Let $M$ and $N$ be closed, orientable, smooth $4$-manifolds that are homotopy equivalent and have torsion-free fundamental group. Stable diffeomorphism conjecture. Then $M$ and $N$…

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…

Fix a prime $p$, and for a group or space $X$ let $\beta_i^p(X)$ denote the dimension of $H_i(X,\mathbf{Z}_p)$. A homology 4-sphere is a 4-manifold with the homology of the 4-spher…