Milnor invariants for spatial graphs

Link-homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component-homotopy, which reduces to link-homotopy in the classical case. Unlike previous attempts at generalizing link-homotopy to spatial graphs, our new relation allows analogues of some standard link-homotopy results and invariants. ticular we can define a type of Milnor group for a spatial graph under component-homotopy, and this group determines whether or not the spatial graph is splittable. More surprisingly, we will also show that whether the spatial graph is splittable up to component-homotopy depends only on the link-homotopy class of the links contained within it. Numerical invariants of the relation will also be produced.