  Given a group $G$ and a class of manifolds $\CC$ (e.g. symplectic, contact,
K\"ahler 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 $\pi_1(M_G)=G$ and
$\pi_i(M_G)=0$ 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 "softness" or
"hardness" \`a 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 "softness" 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, $ht_\CC(G)$ provides a numerical
invariant for finitely presented groups. We give explicit computations of these
invariants for some standard finitely presented groups.
  We use the notion of homotopical height within the "hard" category of
K\"ahler groups to obtain partial answers to questions of Toledo regarding
second cohomology and second group cohomology of K\"ahler groups. We also
modify and generalize a construction due to Dimca, Papadima and Suciu to give a
potentially large class of projective groups violating property FP.
