## Spanier–Whitehead duality in the $K(2)$-local category at $p=2$

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- by Irina Bobkova
- Proc. Amer. Math. Soc.
**148**(2020), 5421-5436 - DOI: https://doi.org/10.1090/proc/15078
- Published electronically: September 4, 2020
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## Abstract:

The fixed point spectra of Morava $E$-theory $E_n$ under the action of finite subgroups of the Morava stabilizer group $\mathbb {G}_n$, and their $K(n)$-local Spanier–Whitehead duals can be used to approximate the $K(n)$-local sphere in certain cases. For any finite subgroup $F$ of $\mathbb {G}_2$ at $p=2$ we prove that the $K(2)$-local Spanier–Whitehead dual of the spectrum $E_2^{hF}$ is $\Sigma ^{44}E_2^{hF}$. These results are analogous to the known results at height 2 and $p=3$. The main computational tool we use is the topological duality resolution spectral sequence for the spectrum $E_2^{h\mathbb {S}_2^1}$ at $p=2$.## References

- Tilman Bauer,
*Computation of the homotopy of the spectrum tmf*, Groups, homotopy and configuration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, pp. 11–40. MR**2508200**, DOI 10.2140/gtm.2008.13.11 - Mark Behrens and Daniel G. Davis,
*The homotopy fixed point spectra of profinite Galois extensions*, Trans. Amer. Math. Soc.**362**(2010), no. 9, 4983–5042. MR**2645058**, DOI 10.1090/S0002-9947-10-05154-8 - Agnès Beaudry,
*The algebraic duality resolution at $p=2$*, Algebr. Geom. Topol.**15**(2015), no. 6, 3653–3705. MR**3450774**, DOI 10.2140/agt.2015.15.3653 - Mark Behrens,
*A modular description of the $K(2)$-local sphere at the prime 3*, Topology**45**(2006), no. 2, 343–402. MR**2193339**, DOI 10.1016/j.top.2005.08.005 - Irina Bobkova and Paul G. Goerss,
*Topological resolutions in $K(2)$-local homotopy theory at the prime 2*, J. Topol.**11**(2018), no. 4, 918–957. MR**3989433**, DOI 10.1112/topo.12076 - A. K. Bousfield,
*The localization of spectra with respect to homology*, Topology**18**(1979), no. 4, 257–281. MR**551009**, DOI 10.1016/0040-9383(79)90018-1 - Ethan S. Devinatz and Michael J. Hopkins,
*Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups*, Topology**43**(2004), no. 1, 1–47. MR**2030586**, DOI 10.1016/S0040-9383(03)00029-6 - P. G. Goerss and M. J. Hopkins,
*Moduli spaces of commutative ring spectra*, Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151–200. MR**2125040**, DOI 10.1017/CBO9780511529955.009 - P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk,
*A resolution of the $K(2)$-local sphere at the prime 3*, Ann. of Math. (2)**162**(2005), no. 2, 777–822. MR**2183282**, DOI 10.4007/annals.2005.162.777 - Hans-Werner Henn,
*The centralizer resolution of the $K(2)$-local sphere at the prime 2*, available from https://hal.archives-ouvertes.fr/hal-01697478/document. - Hans-Werner Henn,
*On finite resolutions of $K(n)$-local spheres*, Elliptic cohomology, London Math. Soc. Lecture Note Ser., vol. 342, Cambridge Univ. Press, Cambridge, 2007, pp. 122–169. MR**2330511**, DOI 10.1017/CBO9780511721489.008 - Thomas Hewett,
*Finite subgroups of division algebras over local fields*, J. Algebra**173**(1995), no. 3, 518–548. MR**1327867**, DOI 10.1006/jabr.1995.1101 - Rebekah Hahn and Stephen Mitchell,
*Iwasawa theory for $K(1)$-local spectra*, Trans. Amer. Math. Soc.**359**(2007), no. 11, 5207–5238. MR**2327028**, DOI 10.1090/S0002-9947-07-04204-3 - Mark Hovey,
*Operations and co-operations in Morava $E$-theory*, Homology Homotopy Appl.**6**(2004), no. 1, 201–236. MR**2076002** - Charles Rezk,
*Supplementary Notes for Math 512*, available from http://www.math.uiuc.edu/~rezk/512-spr2001-notes.pdf. - Charles Rezk,
*Notes on the Hopkins-Miller theorem*, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997) Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 313–366. MR**1642902**, DOI 10.1090/conm/220/03107 - N. P. Strickland,
*Gross-Hopkins duality*, Topology**39**(2000), no. 5, 1021–1033. MR**1763961**, DOI 10.1016/S0040-9383(99)00049-X

## Bibliographic Information

**Irina Bobkova**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 1124377
- ORCID: 0000-0001-8775-2691
- Email: ibobkova@math.tamu.edu
- Received by editor(s): June 24, 2018
- Received by editor(s) in revised form: December 24, 2018, July 2, 2019, and February 24, 2020
- Published electronically: September 4, 2020
- Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, CA, during the Spring 2019 semester, and Grant No. DMS-1638352 and DMS-2005627
- Communicated by: Mark Behrens
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5421-5436 - MSC (2010): Primary 55P25, 55P42, 55T25
- DOI: https://doi.org/10.1090/proc/15078
- MathSciNet review: 4163853