This appendix is dedicated to the body of results about structured ring spectra which are relevant to the String–orientation of elliptic spectra. We include four main topics: a discussion of the stable homotopy theoretic analog of character theory; the compatibility of the results of the preceding chapter on orientations in the homotopy category with power operations for Lubin–Tate spectra; the sheaf of ring spectra enriching the moduli stack of elliptic curves and the spectrum of topological modular forms; and orientations of the spectrum of topological modular forms by maps of structured rings.

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.

Equivariant homotopy theory started from geometrically motivated questions about symmetries of manifolds. Several important equivariant phenomena occur not just for a particular group, but in a uniform way for all groups. Prominent examples include stable homotopy, K-theory or bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., universal symmetries encoded by simultaneous and compatible actions of all compact Lie groups. This book introduces graduate students and researchers to global equivariant homotopy theory. The framework is based on the new notion of global equivalences for orthogonal spectra, a much finer notion of equivalence than what is traditionally considered. The treatment is largely self-contained and contains many examples, making it suitable as a textbook for an advanced graduate class. At the same time, the book is a comprehensive research monograph with detailed calculations that reveal the intrinsic beauty of global equivariant phenomena.

Equivariant homotopy theory started from geometrically motivated questions about symmetries of manifolds. Several important equivariant phenomena occur not just for a particular group, but in a uniform way for all groups. Prominent examples include stable homotopy, K-theory or bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., universal symmetries encoded by simultaneous and compatible actions of all compact Lie groups. This book introduces graduate students and researchers to global equivariant homotopy theory. The framework is based on the new notion of global equivalences for orthogonal spectra, a much finer notion of equivalence than what is traditionally considered. The treatment is largely self-contained and contains many examples, making it suitable as a textbook for an advanced graduate class. At the same time, the book is a comprehensive research monograph with detailed calculations that reveal the intrinsic beauty of global equivariant phenomena.