Complex cobordism is a concept from algebraic topology, a branch of mathematics that studies topological spaces with the methods of abstract algebra. Specifically, complex cobordism is concerned with the relationships between different manifolds (smooth, differentiable structures) via a kind of equivalence that is defined through the notion of cobordism.
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Complex cobordism is the generalized cohomology theory represented by the stable Thom spaces of universal complex vector bundles. Its geometric cycles use maps with chosen stable complex normal data. Its complex Thom classes give Chern classes in complex cobordism, and proper oriented maps have a Gysin map in complex cobordism.