The Smooth 4-dimensional Poincaré Conjecture (S4PC) asserts that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere $S^4$. We prove a special case of the S4PC: If $X$ is a homotopy 4-sphere that can be built with two 2-handles and two 3-handles, and such that one component of the 2-handle attaching link $L$ is a generalized square knot $T(p,q) \# T(-p,q)$, then $X$ is diffeomorphic to $S^4$. This is joint work with Jeffrey Meier.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/9096