We study Vassiliev invariants of links in a 3-manifold M by using chord diagrams labeled by elements of the fundamental group of M. We construct universal Vassiliev invariants of links in M, where M=P 2×[0,1] is a cylinder over the real projective plane P 2, M=Σ×[0,1] is a cylinder over a surface Σ with boundary, and M=S 1×S 2. A finite covering p:N→M induces a map π 1(p)* between labeled chord diagrams that corresponds to taking the preimage p -1(L)⊂N of a link L⊂M. The maps p -1 and π 1(p)* intertwine the constructed universal Vassiliev invariants.
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