  Non-commutative versions of Arveson's curvature invariant and Euler
characteristic for a commuting $n$-tuple of operators are introduced. The
non-commutative curvature invariant is sensitive enough to determine if an
$n$-tuple is free. In general both invariants can be thought of as measuring
the freeness or curvature of an $n$-tuple. The connection with dilation theory
provides motivation and exhibits relationships between the invariants. A new
class of examples is used to illustrate the differences encountered in the
non-commutative setting and obtain information on the ranges of the invariants.
The curvature invariant is also shown to be upper semi-continuous.
