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

Irina Bobkova, Jack Carlisle, Emmett Fitz, Mattie Ji, Peter Kilway, Hillary Kim, Kolton O'Neal, Jacob Schuckman, Scotty Tilton

TL;DR

This work computes the $E_2^{hC_6}$-homology of $\mathbb{RP}^2$ and $\mathbb{RP}^2\wedge\mathbb{C}P^2$ at the prime $2$ by exploiting the Morava $E$-theory height-2 fixed-point framework. The authors reduce the problem to the HFPSS for $E_2^{hC_6}\wedge V(0)$ and $E_2^{hC_6}\wedge Y$, obtained from $E^{hC_2}$ via $C_3$-fixed points and cofiber/sequences, and then determine the differentials $d_3$ and $d_7$ along with extension data. They produce explicit $E_ abla$-pages with 16- and 48-periodic structures, identify key generators such as $x$, $v_1$, $v_2$, $y$, and $\overline{\kappa}$, and show how the cofiber relations control the homotopy groups of the target spectra. The results enhance chromatic understanding of $E_2$-local fixed-point spectra at height $2$ and provide concrete computations for the homology of projective spaces and their smash with complex projective space, informing broader analyses of $L_{K(n)}\mathbb{S}$ via the Devinatz–Hopkins framework.

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.

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 work computes the -homology of and at the prime by exploiting the Morava -theory height-2 fixed-point framework. The authors reduce the problem to the HFPSS for and , obtained from via -fixed points and cofiber/sequences, and then determine the differentials and along with extension data. They produce explicit -pages with 16- and 48-periodic structures, identify key generators such as , , , , and , and show how the cofiber relations control the homotopy groups of the target spectra. The results enhance chromatic understanding of -local fixed-point spectra at height and provide concrete computations for the homology of projective spaces and their smash with complex projective space, informing broader analyses of via the Devinatz–Hopkins framework.

Abstract

Let be the Morava E-theory of height 2 at the prime 2. In this paper, we compute the homotopy groups of and using the homotopy fixed point spectral sequences.

Paper Structure

This paper contains 16 sections, 18 theorems, 45 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The homotopy fixed point spectral sequence for E2hC2
  3. The homotopy fixed point spectral sequence for E2hC2 sm V(0)
  4. The d3-differentials
  5. The E infinity page
  6. Extension Problems
  7. The homotopy fixed point spectral sequence for E2hC6
  8. Computing the E2-page
  9. Computing the intermediate pages
  10. The homotopy fixed point spectral sequence for E2hC6 sm V(0)
  11. The $E_2$ page.
  12. Differentials
  13. The homotopy fixed point spectral sequence for pi_*(C6 sm Y)
  14. The $E_2$ page
  15. $d_3$ and $d_5$ differentials
  16. ...and 1 more sections

Key Result

Lemma 2.1

The $d_3$ differentials in eq:C2-SS are generated by and linearity with respect to $\alpha$, $u_1$ and $u^{\pm 4}$.

Figures (14)

  • Figure 1: The $E_3$ (top) and $E_7$ (bottom) page of the HFPSS for $\operatorname{E}^{hC_2}$. The notation is as follows: $\mathbin{ \ooalign{$$\m@th\bigcirc$$\cr\hidewidth$∙$\hidewidth\cr}}=\mathbb{F}_4\llbracket u_1\rrbracket$, $\bullet=\mathbb{F}_4$, and $\Box=\mathbb{W}\llbracket u_1 \rrbracket$.
  • Figure 2: The $E_8 = E_{\infty}$ page of the HFPSS for $\operatorname{E}^{hC_2}$. The notation is as follows: $\mathbin{ \ooalign{$$\m@th\bigcirc$$\cr\hidewidth$∙$\hidewidth\cr}}=\mathbb{F}_4\llbracket u_1\rrbracket$, $\bullet=\mathbb{F}_4$, $\circ=u_1 \mathbb{F}_4\llbracket u_1 \rrbracket$, $\Box=\mathbb{W}\llbracket u_1 \rrbracket$, and the $\boxtimes$ at $(8,0)$ denotes $2u^{-4}W\oplus u^{-4}u_1W\llbracket u_1\rrbracket$. The homotopy groups are 16-periodic, with periodicity generator $u^{-8}$.
  • Figure 3: The $E_3$ page of the HFPSS for $\operatorname{E}^{hC_2} \wedge V(0)$. The symbol $\mathbin{ \ooalign{$$\m@th\bigcirc$$\cr\hidewidth$∙$\hidewidth\cr}}$ represents $\mathbb{F}_4\llbracket u_1\rrbracket$.
  • Figure 4: The $E_7$ page of the HFPSS for $\operatorname{E}^{hC_2} \wedge V(0)$. The symbol $\bullet$ represents $\mathbb{F}_4$, and $\mathbin{ \ooalign{$$\m@th\bigcirc$$\cr\hidewidth$∙$\hidewidth\cr}}$ represents $\mathbb{F}_4\llbracket u_1\rrbracket$.
  • Figure 5: The homotopy groups of $\operatorname{E}^{hC_2} \wedge V(0)$. The notation is: $\bullet =\mathbb{F}_4$, and $\mathbin{ \ooalign{$$\m@th\bigcirc$$\cr\hidewidth$∙$\hidewidth\cr}} = \mathbb{F}_4\llbracket u_1\rrbracket$. The lines of slope 1 indicate multiplication by $\eta$. The lines connecting elements in the same stem indicate group extensions. The homotopy groups are 16-periodic.
  • ...and 9 more figures

Theorems & Definitions (29)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1: Section 2.2 of 1a7d99963b1048d8ad18fe824397caab
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Lemma 3.6
  • ...and 19 more