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Two links are called link-homotopic if they are transformed to each other bya sequence of self-crossing changes and ambient isotopies. The notion oflink-homotopy is generalized to spatial graphs and it is calledcomponent-homotopy. The link-homotopy classes were classified by Habegger andLin through the classification of the link-homotopy classes of string links. Inthis paper, we classify colored string links up to colored link-homotopy byusing the Habegger-Lin theory. Moreover, we classify colored links and spatialgraphs up to colored link-homotopy and component-homotopy respectively.

Homotopy theories of colored links and spatial graphs