The $\operatorname{E}_2^{hC_6}$-homology of $\mathbb{R}P^2$ and $\mathbb{R}P^2 \wedge \mathbb{C}P^2$

TL;DR

This paper computes the homotopy groups of certain Morava E-theory spectra related to real and complex projective spaces at height 2 and prime 2, advancing understanding of their equivariant homotopy properties.

Contribution

It provides explicit calculations of homotopy groups for Morava E-theory fixed points on real and complex projective spaces, a novel contribution in chromatic homotopy theory.

Findings

Homotopy groups of E_2^{hC_6}   ext{RP}^2 computed.

Homotopy groups of E_2^{hC_6}   ext{RP}^2  ext{CP}^2 computed.

Results obtained via homotopy fixed point spectral sequences.

Abstract

Let $\operatorname{E}_2$ be the Morava E-theory of height 2 at the prime 2. In this paper, we compute the homotopy groups of $\operatorname{E}_2^{hC_6} \wedge \mathbb{R}P^2$ and $\operatorname{E}_2^{hC_6} \wedge \mathbb{R}P^2 \wedge \mathbb{C}P^2$ using the homotopy fixed point spectral sequences.