6 problems

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-…

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…

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…

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…