Some extensions in the Adams spectral sequence and the 51–stem

Abstract

We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of π 51 is Z/8 ⊕ Z/8 ⊕ Z/2. This was the last unsolved 2-extension problem left by the recent works of Isaksen and the authors ([5], [7], [20]) through the 61-stem.The proof of this result uses the RP ∞ technique, which was introduced by the authors in [20] to prove π 61 = 0. This paper advertises this technique through examples that have simpler proofs than in [20].Proposition 1.1. There is a nontrivial 2-extension from h 0 h 3 g 2 to gn in the 51-stem. the 51-stem and some extensionsWe first establish the following lemma. Lemma 2.1. In the Adams E 2 -page, we have the following Massey products in the 46-stem: gn = N, h 1 , h 2 = N, h 2 , h 1 Proof. By Bruner's computation [4], there is a relation in bidegree (t − s, s) = (81, 15): gnr = mN.

Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

“…where 𝛽 ∈ 𝜋 2 = ℤ/2 generated by 𝜂 2 , and 𝛼 ∈ 𝜋 10 = ℤ/2 generated by {𝑃ℎ 2 1 }. For a precise argument of this fact, we refer to Lemma 5.3 of [WX18].…”

The topological modular forms of RP2$\mathbb {R}P^2$ and RP2∧CP2$\mathbb {R}P^2 \wedge \mathbb {C}P^2$

“…𝑎 [35,3] = Δ𝜈 2 𝑎[5, 1] 𝑎 [36,2] = Δ𝑎 [12,2] 𝑎 [38,2] = Δ𝑣 1 𝑎 [12,2] 𝑎 [41,3] = Δ𝑎 [17,3] 𝑎[43, 3] = Δ𝑣 1 𝑎 [17,3] 𝑎[45, 3] = Δ𝑣 2 1 𝑎 [17,3] All κ-free families at 𝐸 9 are given by these classes and their Δ 2 -multiples. All the elements in filtrations four and above are κ-multiples of these generators.…”

“…Proof of Lemma In our Atiyah–Hirzebruch notation, we can rewrite the two $$\nu$$‐extensions of Lemma 6.39 as (1)$$\nu \cdot \bar{\kappa }^2 \kappa [3] = \Delta \eta \kappa \bar{\kappa }[1]$$, (2)$$\nu \cdot \bar{\kappa }^3 [2] = \Delta ^2 \nu \kappa [0]$$. We first prove (2), namely, that $$\nu \cdot \bar{\kappa }^3 [2] = \Delta ^2 \nu \kappa [0]$$. In $$\pi _*tmf \wedge C_\eta$$, we have $$\begin{equation*} \nu \cdot \bar{\kappa }^3 [2] = \langle \nu , \bar{\kappa }^3, \eta \rangle [0] \end{equation*}$$by [41, Lemma 5.3]. By Moss's theorem and the differential $$d_{11}(\Delta ^2 \kappa ) = \eta \bar{\kappa }^3$$ in the elliptic spectral sequence of …”

“…In 𝜋 * 𝑡𝑚𝑓 ∧ 𝐶 𝜂 , we have 𝜈 ⋅ κ3 [2] = ⟨𝜈, κ3 , 𝜂⟩[0]by[41, Lemma 5.3]. By Moss's theorem and the differential 𝑑 11 (Δ 2 𝜅) = 𝜂 κ3 in the elliptic spectral sequence of 𝑡𝑚𝑓, we have ⟨𝜈, κ3 , 𝜂⟩ = Δ 2 𝜈𝜅.Mapping this relation along the inclusion 𝐶 𝜂 → 𝑌 gives us (2).For(1), note that in 𝜋 * 𝑡𝑚𝑓 ∧ Σ𝐶 𝜂 , we have𝜈 ⋅ κ2 𝜅[3] = ⟨𝜈, κ2 𝜅, 𝜂⟩[1]by[41, Lemma 5.3]. Since κ2 𝜅 is 𝜈-divisible in 𝜋 * 𝑡𝑚𝑓, we may shuffle ⟨𝜈, κ2 𝜅, 𝜂⟩ = ⟨ κ2 𝜅, 𝜈, 𝜂⟩.By Moss's theorem and the differential 𝑑 5 (Δ𝜅 κ) = 𝜈 κ2 𝜅 in 𝑡𝑚𝑓, we have ⟨ κ2 𝜅, 𝜈, 𝜂⟩ = Δ𝜂𝜅 κ.Pulling back this relation along the quotient map 𝑌 → Σ𝐶 𝜂 gives (1).…”

Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant

“…where β ∈ π 2 = Z/2 generated by η 2 , and α ∈ π 10 = Z/2 generated by {P h 2 1 }. For a precise argument of this fact, we refer to Lemma 5.3 of [WX18].…”