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On cyclic fixed points of spectra

Mathematische Zeitschrift volume 276, pages 81–91 (2014)Cite this article

Abstract

For a finite \(p\)-group \(G\) and a bounded below \(G\)-spectrum \(X\) of finite type mod \(p\), the \(G\)-equivariant Segal conjecture for \(X\) asserts that the canonical map \(X^G \rightarrow X^{hG}\), from \(G\)-fixed points to \(G\)-homotopy fixed points, is a \(p\)-adic equivalence. Let \(C_{p^n}\) be the cyclic group of order \(p^n\). We show that if the \(C_p\)-equivariant Segal conjecture holds for a \(C_{p^n}\)-spectrum \(X\), as well as for each of its geometric fixed point spectra \(\varPhi ^{C_{p^e}}(X)\) for \(0 < e < n\), then the \(C_{p^n}\)-equivariant Segal conjecture holds for \(X\). Similar results also hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Aarhus University, Aarhus, Denmark

    Marcel Bökstedt

  2. Department of Mathematics, Wayne State University, Detroit, MI, USA

    Robert R. Bruner

  3. Department of Mathematics, University of Oslo, Oslo, Norway

    Sverre Lunøe-Nielsen & John Rognes

Authors
  1. Marcel Bökstedt
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  2. Robert R. Bruner
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  3. Sverre Lunøe-Nielsen
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  4. John Rognes
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Correspondence to John Rognes.

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Bökstedt, M., Bruner, R.R., Lunøe-Nielsen, S. et al. On cyclic fixed points of spectra. Math. Z. 276, 81–91 (2014). https://doi.org/10.1007/s00209-013-1187-0

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  • DOI: https://doi.org/10.1007/s00209-013-1187-0

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