10 problems

Let $E(n)$ be the simply connected elliptic surface, and let $E(1)_{2,3}$ denote the result of applying $2$- and $3$-log transforms to $E(1)$; this is an exotic…

Let $E(n)$ be the simply connected elliptic surface, and let $E(n)_{p,q}$ be the complex surface obtained from $E(n)$ by a pair of $p$- and $q$-log transforms, where $n\geq 1$,…

A smooth 4-manifold is said to admit a handle decomposition without 1- or 3-handles if it has a handle decomposition containing no handles of index 1 or 3. Handle decomposition con…

Let $M$ be a compact contractible $4$-manifold whose boundary is a homology sphere. Write $\operatorname{int}(M)$ for its interior, and let g.s.c. denote geometric simple connectiv…

Unbounded handle complexity conjecture. Suppose $\Gamma_Y(i)<i/2$. Then the minimum number of $1$-handles and $2$-handles among four-manifolds that fill $\#_nY\#_n-Y$ grows without…

Let $Y_n=\#_n\Sigma(2,3,5)$, where $\Sigma(2,3,5)$ is the Poincaré homology sphere. A bounding four-manifold for $Y_n\#-Y_n$ is a four-manifold whose boundary is $Y_n\#-Y_n$. Unbou…

Let $M$ be a closed simply-connected PL $4$-manifold. A contracted triangulation is a triangulation satisfying the hypothesis of Theorem (b). Contracted-triangulation conjecture. E…

No 1-handles conjecture. Every simply-connected 4-manifold has a handle decomposition with no 1-handles.

Kirby's conjecture. Every simply connected closed smooth $4$-manifold is geometrically $1$-connected.

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…