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Homotopical Height

Given a group $G$ and a class of manifolds $\CC$ (e.g. symplectic, contact, Kähler etc), it is an old problem to find a manifold $M_G \in \CC$ whose fundamental group is $G$. This article refines it: for a group $G$ and a positive integer $r$ find $M_G \in \CC$ such that $π_1(M_G)=G$ and $π_i(M_G)=0$ for $1<i<r$. We thus provide a unified point of view systematizing known and new results in this direction for various different classes of manifolds. The largest $r$ for which such an $M_G \in \CC$ can be found is called the homotopical height $ht_\CC(G)$. Homotopical height provides a dimensional obstruction to finding a $K(G,1)$ space within the given class $\CC$, leading to a hierarchy of these classes in terms of &#34;softness&#34; or &#34;hardness&#34; à la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes $\SP$ and $\CA$ of closed symplectic and complex manifolds exhibit intermediate &#34;softness&#34; in the sense that every finitely presented group $G$ can be realized as the fundamental group of a manifold in $\SP$ and a manifold in $\CA$. For these classes, $ht_\CC(G)$ prov