25 problems

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

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 $0$.

Let $N$ be a smooth manifold, and consider the Almost Embeddings Theorem (b), the cited results corresponding to [Sk02, Theorem 2.3.q], and the injectivity assertion corresponding…

Let $M$ be an elliptic $3$-manifold, and let $W$ be a smooth $s$-cobordism from $M$ to itself. Let $\widetilde{W}$ be its universal cover. Chen's smooth product conjecture. $W$ is…

Let $X$ be a closed smooth spin topological four-manifold. For an appropriately chosen basis of its second homology, suppose its intersection matrix has the form … where $E_8$ is t…

Let $X$ be a simply connected closed smooth spin $4$-manifold, and write its even unimodular intersection form as … where $E_8$ is the positive-definite even form of rank $8$ and…

Let $X$ be a homotopy $\mathbb{CP}^2$ constructed in Proposition from a Berge knot. Standardness conjecture. The manifold $X$ is diffeomorphic to the standard $\mathbb{CP}^2$. Thes…

Let … The set $\mathcal{C}\mathcal{S}$ parametrizes standard Cappell–Shaneson matrices via $(c,d,n)\mapsto X_{c,d,n}$. Let $\sim$ denote Gompf equivalence on…

11/8 conjecture. A smooth structure on $X$ requires

Smooth framed isotopy of tame smooth tangles implies diffeomorphism. Given smooth $m$-tame tangles $f:W\to\mathbb{I}^n$ and $f':W'\to\mathbb{I}^n$, if $f$ and $f'$ are smoothly fra…

Framed homeomorphism of tame smooth tangles implies diffeomorphism. If two tame smooth tangles $W\to\mathbb{I}^n$ and $W'\to\mathbb{I}^n$ are framed stratified homeomorphic, then…

Unknotting conjecture. If a $2$-knot has infinite cyclic knot group, then it is unknotted.

Let $F$ be an embedded $2$-sphere in $S^4$. The 4-dimensional unknotting conjecture. The sphere $F$ bounds a $4$-ball if and only if the fundamental group of its complement is infi…

Let $X$ and $Y$ be smooth CW complexes, and let $f:Y\to X$ be a continuous map. A subset $A\subset Y$ is closed, and $f$ is smooth on $A$ when this condition is relevant. Whitney a…

A smooth 3-sphere embedded in the smooth 4-sphere is a smooth separating hypersurface. Four-dimensional smooth Schoenflies conjecture. Every smooth 3-sphere in the 4-sphere is smoo…

Let $W$ be a compact smooth $4$-manifold with boundary. The manifold $W$ is spineless in the sense relevant here if it is homotopy equivalent to $S^2$ but has no embedding…

Let $M$ be a closed oriented smooth $4$-manifold with indefinite even intersection form. Write $b_2(M)$ for its second Betti number and $\operatorname{sign}(M)$ for its signature.…

A smooth embedding of $S^3$ into $\mathbb{R}^4$ is a smooth embedding of the $3$-sphere in Euclidean $4$-space, and the standard round sphere is the usual embedding of $S^3$ as the…

Small perturbation Seiberg–Witten conjecture. Under these assumptions, $M$ satisfies property $(\ast)$.

Let a surface in a 4-manifold be specified by its homology class and genus. Conjecture. Such simple criteria are never sufficient to determine the surface up to isotopy in the smoo…

Linearisation conjecture. Every smooth finite group action on $\mathbb S^4$ with an isolated fixed point is smoothly conjugate to a linear action.

Let $L=L_1\cup L_2$ satisfy the hypotheses of the preceding Kirby-calculus formulation, but impose no restriction on the number of components of $L_2$. If $p+q\ne0$, then $L$ can b…

Let $L=L_1\cup L_2$ satisfy the hypotheses of the preceding Kirby-calculus formulation, except that $L_2$ is required only to generate $H_1(S^3-L_1;\mathbb Z)$ rather than…

Let $L=L_1\cup L_2$ be a link in $S^3$, with $L_1$ a dotted $p$-component unlink and $L_2$ a framed link of $p+q$ components. Suppose that $L_2$ normally generates $\pi_1(S^3-L_1)$…