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We exhaustively classify topological equivariant complex vector bundles overtwo-torus under a compact Lie group (not necessarily effective) action. It isshown that inequivariant Chern classes and isotropy representations at (atmost) six points are sufficient to classify equivariant vector bundles except afew cases. To do it, we calculate homotopy of the set of equivariant clutchingmaps. And, classification on real projective plane, Klein bottle will appearsoon

Classification of equivariant vector bundles over two-torus