  Enumerative geometry studies the intersection theory of virtual fundamental
classes in the homology of moduli spaces. These usually depend on auxiliary
parameters and are then related by wall-crossing formulas.
  We construct a homological graded Lie bracket on the homology of moduli
spaces which can be used to express universal wall-crossing formulas. For this
we develop a new topological theory of pushforward operations for principal
bundles with orientations in twisted K-theory. We prove that all rational
pushforward operations are obtained from a single new projective Euler
operation, which also has applications to Chern classes in twisted K-theory.
