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.
In particular 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.Comment: 11 pages, 5 figure
