162 problems

Does there exist an open $4$-manifold $M$ that admits a finite-dimensional compact ANR $\mathcal{Z}$-compactification but is not pseudo-collarable?

A homotopy 4-sphere is a smooth 4-manifold homotopy equivalent to the 4-sphere $S^4$. Smooth 4-dimensional Poincaré conjecture. Every homotopy 4-sphere is diffeomorphic to $S^4$. T…

Smooth Poincaré conjecture in dimension 4. There are no exotic $S^4$'s; equivalently,

Let $M^n_1$ and $M^n_2$ be two closed orientable $n$-manifolds. The regular genus additivity conjecture. … Regular genus is known to be subadditive under connected sum, and the equ…

Let $X$ be a smooth simply connected $4$-manifold. For a class $xi\in H^2(X,\mathbb Z)$ defining a wall, an integer $N$, and the associated wall-crossing term $\delta^X_{\xi,N}$, l…

A smoothly embedded projective plane in $S^4$ is knotted if it is not the standard unknotted embedding, and it is reducible if it admits an unknotted summand that is not a $2$-sphe…

Let $X$ be a closed, oriented, aspherical $4$-manifold, and let $\tau(X)$ denote its signature. Gromov–Lück inequality. One has … This inequality is a refined form of the Hopf prob…

A Cappell–Shaneson matrix is a matrix $A\in SL(3;\mathbb{Z})$ satisfying $\det(A-I)=1$. Two such matrices are Gompf equivalent when they are related by Gompf's equivalence relation…

Let $L$ be an R-link, meaning an $n$-component link in $S^3$ whose 0-surgery yields $Y_n=\mathop{\#}^n(S^1\times S^2)$. Let $U$ be a 0-framed unlink. The stable Generalized Propert…

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.