  In this paper we define and study the moduli space of metric-graph-flows in a
manifold M.
  This is a space of smooth maps from a finite graph to M, which, when
restricted to each edge, is a gradient flow line of a smooth (and generically
Morse) function on M. Using the model of Gromov-Witten theory, with this moduli
space replacing the space of stable holomorphic curves in a symplectic
manifold, we obtain invariants, which are (co)homology operations in M. The
invariants obtained in this setting are classical cohomology operations such as
cup product, Steenrod squares, and Stiefel-Whitney classes. We show that these
operations satisfy invariance and gluing properties that fit together to give
the structure of a topological quantum field theory. By considering equivariant
operations with respect to the action of the automorphism group of the graph,
the field theory has more structure. It is analogous to a homological conformal
field theory. In particular we show that classical relations such as the Adem
relations and Cartan formulae are consequences of these field theoretic
properties.
  These operations are defined and studied using two different methods. First,
we use algebraic topological techniques to define appropriate virtual
fundamental classes of these moduli spaces. This allows us to define the
operations via the corresponding intersection numbers of the moduli space.
Secondly, we use geometric and analytic techniques to study the smoothness and
compactness properties of these moduli spaces. This will allow us to define
these operations on the level of Morse-Smale chain complexes, by appropriately
counting metric-graph-flows with particular boundary conditions.
