Periodic phenomena in equivariant stable homotopy theory

Abstract

Building off of many recent advances in the subject by many different researchers, we describe a picture of A-equivariant chromatic homotopy theory which mirrors the now classical non-equivariant picture of Morava, Miller-Ravenel-Wilson, and Devinatz-Hopkins-Smith, where A is a finite abelian p-group. Specifically, we review the structure of the Balmer spectrum of the category of A-spectra, and the work of Hausmann-Meier connecting this to MU_A and equivariant formal group laws. Generalizing work of Bhattacharya-Guillou-Li, we introduce equivariant analogs of v_n-self maps, and generalizing work of Carrick and Balderrama, we introduce equivariant analogs of the chromatic tower, and give equivariant analogs of the smash product and chromatic convergence theorems. The equivariant monochromatic theory is also discussed. We explore computational examples of this theory in the case of A = C_2, where we connect equivariant chromatic theory with redshift phenomena in Mahowald invariants.

Figures (5)

• Figure 2.1: Structure of $\mathop{\mathrm{Spc}}\nolimits(A)$ in the vicinity of $B < C \le A$ with $C/B$ a $p$-group of rank $r$.
• Figure 2.2: The lattice structure of $\mathop{\mathrm{Spc}}\nolimits_{(p)}(C_p)$, and some examples of open and closed subsets.
• Figure 2.3: Closed and open subsets of $\mathrm{Sub}(C_2 \times C_2)$.
• Figure 2.4: Some closed subsets of $\mathrm{Sub}(C_2 \times C_2)$ determined by characters.
• Figure 3.1: Visualization of an equivariant formal group law $(\mathbb{G}, \varphi)$ over $X$. This particular picture is meant to represent a $C_3$-equivariant formal group. The three curved lines represent the image $\mathop{\mathrm{im}}\nolimits \varphi$, and $\mathbb{G}$ is represented by the dotted "neighborhood" of $\mathop{\mathrm{im}}\nolimits \varphi$ in this visualization. The straight curve and the bottom curve represent the closed subsechemes $\varphi_1$ and $\varphi_\alpha$, respectively, where $\alpha \in C_3^\vee$ is a non-trivial character. The dotted "neighborhoods of these two lines represent the formal subschemes $\mathbb{G}_1$ and $\mathbb{G}_\alpha$, respectively.

Theorems & Definitions (153)

• Remark 2.1
• Proposition 2.7
• proof
• Proposition 2.12
• Theorem 2.13: Greenlees-May
• Proposition 2.14: Bauer Bauer
• Lemma 2.15
• proof
• Lemma 2.16
• Corollary 2.17