### Abstract

We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the E_{∞} string orientation of tmf, the spectrum of topological modular forms. Specifically, we show that, for an E∞ ring spectrum A, the classical construction of gl_{1}A, the spectrum of units, is the right adjoint of the functor Σ^{∞}+Ω^{∞}: ho(connective spectra) -→ ho(E_{∞} ring spectra). To a map of spectra f : b -→ bgl_{1}A, we associate an E_{∞} A-algebra Thom spectrum Mf, which admits an E_{∞} A-algebra map to R if and only if the composition b -→ bgl1A -→ bgl_{1}R is null; the classical case developed by May, Quinn, Ray, and Tornehave arises when A is the sphere spectrum. We develop the analogous theory for A_{∞} ring spectra: if A is an A_{∞} ring spectrum, then to a map of spaces f : B -→ BGL_{1}A, we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -→ BGL1A -→ BGL_{1}R is null. Our work is based on a new model of the Thom spectrum as a derived smash product.

Original language | English (US) |
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Article number | jtu009 |

Pages (from-to) | 1077-1117 |

Number of pages | 41 |

Journal | Journal of Topology |

Volume | 7 |

Issue number | 4 |

DOIs | |

State | Published - Jan 21 2013 |

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Journal of Topology*,

*7*(4), 1077-1117. [jtu009]. https://doi.org/10.1112/jtopol/jtu009