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.
In his 1970 MIT notes @cite_23 , In the version available at http: www.maths.ed.ac.uk aar books gtop.pdf , see the note on page 236. Sullivan introduced the classical obstruction theory for orientations and suggested that Dold's theory of homotopy functors @cite_21 could be used to construct the space @math of @math -oriented spherical fibrations. He also mentioned that the technology to construct the delooping @math was on its way. Soon thereafter, May, Quinn, Ray, and Tornehave in @cite_20 constructed the space @math in the case that @math is an @math ring spectrum, and described the associated obstruction theory for orientations of spherical fibrations.
@cite_21 @cite_20 @cite_23
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? functors.- Coordinate-free spectra.- Orientation theory.- E? ring spectra.- On kO-oriented bundle theories.- E? ring spaces and bipermutative categories.- The recognition principle for E? ring spaces.- Algebraic and topological K-theory.- Pairings in infinite loop space theory. Algebraic Constructions.- Homotopy Theoretical Localization.- Completions in Homotopy Theory.- Spherical Fibrations.- Algebraic Geometry.- The Galois Group in Geometric Topology.