14 problems

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 $M$ 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 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…

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

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…

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…

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…

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

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 $\pi_1(S^3-L_1)$…