$n$-knots in $S^n\times S^2$ and contractible $(n+3)$-manifolds

Abstract:

In $1961$, Mazur [Ann. of Math. (2) 73 (1961), pp. 221–228] constructed a contractible, compact, smooth $4$-manifold which is not homeomorphic to the standard $4$-ball, using a $0$-handle, a $1$-handle and a $2$-handle. In this paper, for any integer $n\geq 2$, we construct a contractible, compact, smooth $(n+3)$-manifold which is not homeomorphic to the standard $(n+3)$-ball, using a $0$-handle, an $n$-handle and an $(n+1)$-handle. The key step is the construction of an interesting knotted $n$-sphere in $S^n\times S^2$ generalizing the Mazur pattern. As a corollary, for any integer $n\geq 2$, there exists a smooth involution of $S^{n+3}$ whose fixed point set is not homeomorphic to $S^{n+2}$.