2 - Complex Bordism

Summary

Our goal in this chapter is to prove Quillen’s complex analog of Thom’s calculation of the bordism ring. We approach this using cyclic power operations in bordism homology, giving a brief tour of the general theory of power operations along the way, and ultimately employing Quillen’s comparison formula with stable Landweber–Novikov operations. In order to gain a handle on the stable operations, we investigate the formal schemes associated to classifying spaces for complex vector bundles and the algebro–geometric interpretation of the cohomology of Thom spectra. As we develop the topological results, we begin to re-prove in greater generality the specialized algebraic results from the preceding chapter as we find it necessary.