A simple construction of taut submanifolds

Abstract

We show that any integral second cohomology class of a closed manifold $X^n$, $n\ge 4$, admits, as a Poincaré dual, a submanifold $N$ such that $X\setminus N$ has a handle decomposition with no handles of index bigger than $\left(n+1\right)∕2$. In particular, if $X$ is an almost complex manifold of dimension at least $6$, the complement can be given a structure of a Stein manifold.