Spectral Algebra Models of Unstable \(v_n\)-Periodic Homotopy Theory

  • Mark Behrens
  • Charles RezkEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 309)


We give a survey of a generalization of Quillen–Sullivan rational homotopy theory which gives spectral algebra models of unstable \(v_n\)-periodic homotopy types. In addition to describing and contextualizing our original approach, we sketch two other recent approaches which are of a more conceptual nature, due to Arone-Ching and Heuts. In the process, we also survey many relevant concepts which arise in the study of spectral algebra over operads, including topological André-Quillen cohomology, Koszul duality, and Goodwillie calculus.


\(v_n\)-periodic homtopy theory Bousfield–Kuhn functor Topological André-Quillen cohomology 



The authors benefited greatly from conversations with Greg Arone, Michael Ching, Bill Dwyer, Rosona Eldred, Sam Evans, John Francis, John Harper, Gijs Heuts, Mike Hopkins, Nick Kuhn, Jacob Lurie, Mike Mandell, Akhil Mathew, Anibal Medina, Lennart Meier, Luis Alexandre Pereira, and Yifei Zhu. The authors are grateful to Norihiko Minami for encouraging this submission to these conference proceedings, honoring the memory of Tetsusuke Ohkawa. The authors would also like to thank the referee for his/her many useful comments and corrections. Both authors were supported by grants from the NSF.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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