  The covering type of a space $X$ is defined as the minimal cardinality of a
good cover of a space that is homotopy equivalent to $X$. We derive estimates
for the covering type of $X$ in terms of other invariants of $X$, namely the
ranks of the homology groups, the multiplicative structure of the cohomology
ring and the Lusternik-Schnirelmann category of $X$. By relating the covering
type to the number of vertices of minimal triangulations of complexes and
combinatorial manifolds, we obtain, within a unified framework, several
estimates which are either new or extensions of results that have been
previously obtained by ad hoc combinatorial arguments. Moreover, our methods
give results that are valid for entire homotopy classes of spaces.
