We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the string orientation of tmf, the spectrum of topological modular forms. We also develop the analogous theory of Thom spectra and orientations for associative ring spectra. Our work is based on a new model of the Thom spectrum as a derived smash product. An earlier version of this paper was part of arXiv:0810.4535.
Various aspects of the theory of units and Thom spectra have been revisited by a number of authors as the foundations of stable homotopy theory have advanced. For example, Schlichtkrull @cite_10 studied the units of a symmetric ring spectrum, and May and Sigurdsson @cite_30 have studied units and orientations from the perspective of their categories of parametrized spectra. Recently May has prepared an authoritative paper revisiting operad (ring) spaces and operad (ring) spectra from a modern perspective @cite_7 , which has substantial overlap with some of the review of the classical foundations in .
@cite_30 @cite_10 @cite_7
1822242587 2124463956 2013924646
Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The @math -model structure on @math Equivariant @math -type model structures Ex-fibrations and ex-quasifibrations The equivalence between Ho @math and @math Parametrized equivariant stable homotopy theory: Introduction to Part III Enriched categories and @math -categories The category of orthogonal @math -spectra over @math Model structures for parametrized @math -spectra Adjunctions and compatibility relations Module categories, change of universe, and change of groups Parametrized duality theory: Introduction to Part IV Fiberwise duality and transfer maps Closed symmetric bicategories The closed symmetric bicategory of parametrized spectra Costenoble-Waner duality Fiberwise Costenoble-Waner duality Homology and cohomology, Thom spectra, and addenda: Introduction to Part V Parametrized homology and cohomology theories Equivariant parametrized homology and cohomology Twisted theories and spectral sequences Parametrized FSP's and generalized Thom spectra Epilogue: Cellular philosophy and alternative approaches Bibliography Index Index of notation. Let GL1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition E∞ ring spectra were defined in 1972, but the term has since acquired several alternative meanings. The same is true of several related terms. The new formulations are not always known to be equivalent to the old ones and even when they are, the notion of “equivalence” needs discussion: Quillen equivalent categories can be quite seriously inequivalent. Part of the confusion stems from a gap in the modern resurgence of interest in E∞ structures. E∞ ring spaces were also defined in 1972 and have never been redefined. They were central to the early applications and they tie in implicitly to modern applications. We summarize the relationships between the old notions and various new ones, explaining what is and is not known. We take the opportunity to rework and modernize many of the early results. New proofs and perspectives are sprinkled throughout.