On cyclic fixed points of spectra

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 --> X^{hG} is a p-adic equivalence. Let C_{p^n} be the cyclic group of order p^n. We show that if the C_p Segal conjecture holds for a C_{p^n} spectrum X, as well as for each of its C_{p^e} geometric fixed points for 0 < e < n, then then C_{p^n} Segal conjecture holds for X. Similar results 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.

References in corpus (1)

• The Segal conjecture for topological Hochschild homology of complex cobordism

Cited by in corpus (7)

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• On $K(1)$-local $\mathrm{TR}$
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• The Segal Conjecture for topological Hochschild homology of the Ravenel spectra