Abstract:Bivariant theories and the Grothendieck group of 1-cycles.  I will recall bivariant theories, which assign to a map X-->Y a group capturing the cohomology of Y with coefficients in fiberwise homology of X-->Y (or the reverse).  Then I will show how to define the (very tautological) Grothendieck group of 1-cycles in complex threefolds based on these groups.  This group classifies curve enumeration problems with insertions, including family (in particular, equivariant) enumeration problems.For example, it contains equivariant local curve elements x_{g,m,k} which represent the S^1-equivariant enumerative problem of 1-cycles of degree m inside a three-dimensional local curve E of genus g and chern number k.