Survey on Cappell-Shaneson homotopy 4-sphere

Abstract

Cappell and Shaneson constructed a family of homotopy 4-spheres in 1976, and some of them are double covers of an exotic $\mathbb{R}P^4$. They are well-known potential counterexamples for the 4-dimensional smooth Poincaré conjecture. The manifolds are parametrized by a matrix $A\in SL(\mathbb{Z}, 3)$ such that $\det (I-A)=-1$, and $\epsilon\in\mathbb{Z}_2$, which represents a framing of gluing map of $S^2\times D^2$. In this paper, we focus on $\Sigma_m^{\epsilon}$ constructed by smaller family $A_m$ depending only on an integer $m$. In fact, every $A_m^\epsilon$ turns out to be diffeomorphic to the standard sphere $S^4$. This paper is based on papers of Cappell, Shaneson, Akbulut, Kirby, Aitchison, Rubinstein, and Gompf, and follows the proofs in historical order as a survey paper with no author's own result. For the proof of Gompf, in particular, we give more details with handle calculus and isotopy.