## Tbilisi Mathematical Journal

### Higher chromatic analogues of twisted $K$-theory

Mehdi Khorami

#### Abstract

In this paper we introduce a new family of twisted $K(n)$-local homology theories. These theories are given by the spectra $R_n= E_n^{hS\mathbb G_n}$, twisted by a class $H\in H^{n+2}(X, \mathbb Z_p)$. Here $E_n^{hS\mathbb G_n}$ are the homotopy fixed point spectra under the action of the subgroup $S\mathbb G_n$ of the Morava stabilizer group where $S\mathbb G_n$ is the kernel of the determinant homomorphism $\text{det}:\mathbb G_n\to \mathbb Z_p^\times$. These spectra were utilized in [8] by C. Westerland to study higher chromatic analogues of the J-homomorphism. We investigate some of the properties of these new twisted theories and discuss why we consider them as a generalization of twisted $K$-theory to higher chromatic levels.

#### Article information

Source
Tbilisi Math. J., Volume 12, Issue 2 (2019), 153-162.

Dates
Accepted: 27 April 2019
First available in Project Euclid: 21 June 2019

https://projecteuclid.org/euclid.tbilisi/1561082574

Digital Object Identifier
doi:10.32513/tbilisi/1561082574

Mathematical Reviews number (MathSciNet)
MR3973266

#### Citation

Khorami, Mehdi. Higher chromatic analogues of twisted $K$-theory. Tbilisi Math. J. 12 (2019), no. 2, 153--162. doi:10.32513/tbilisi/1561082574. https://projecteuclid.org/euclid.tbilisi/1561082574

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