Mod-two cohomology rings of alternating groups

Proposition A.1.

The product γ 2 , m 2 is represented by the Fox–Neuwirth cochain α 2 , m ⁢ ( 2 ) with m blocks of three repeated 2’s.

The product γ ℓ , m ⋅ γ 1 , m ⁢ 2 ℓ - 1 is represented by the Fox–Neuwirth cochain α ℓ , m ⁢ ( 1.5 ) , which has m blocks of the form [ 2 , 1 , 2 , … , 1 , 2 ] , each of length 2 ℓ - 1 .

Proof.

We perform the arguments in parallel. We start with m = 1 , showing that both the product γ 2 , 1 2 and [ 2 , 2 , 2 ] are linear dual in the Dyer–Lashof basis of q 2 , 2 (respectively, both γ ℓ , 1 ⋅ γ 1 , 2 ℓ - 1 and [ 2 , 1 , 2 , … , 1 , 2 ] are the linear dual of q 1 , … , 1 , 2 ). We calculate pairings, starting with γ 2 , 1 2 , whose value on some x is equal to that of γ 2 , 1 ⊗ γ 2 , 1 (respectively, γ ℓ , 1 ⊗ γ 1 , 2 ℓ - 1 ) on Δ ⁢ x . For x = q 2 , 2 (respectively, q 1 , … , 1 , 2 ) we see that the only term in Δ ⁢ q 2 , 2 in the correct pair of degrees is q 1 , 1 ⊗ q 1 , 1 (respectively, q 1 , … , 1 ⊗ q 0 , … , 0 , 1 ), which pairs to one by definition. Any *-decomposable will have decomposable coproduct, and these will evaluate to zero on a tensor factor of γ ℓ , 1 by definition.

For pairings with [ 2 , 2 , 2 ] (respectively, [ 2 , 1 , 2 , … , 1 , 2 ] ), * -decomposable classes also evaluate to zero, by the analogue of Theorem 4.16. We then apply the symmetric groups version of Proposition 4.15 to evaluate [ 2 , 2 , 2 ] on q 2 , 2 , which by definition is represented by a map from S 2 × 𝒮 2 ( ℝ ⁢ P 2 × ℝ ⁢ P 2 ) to Conf ¯ 4 ⁢ ( ℝ ∞ ) sending ( u , v , w ) / ∼ to the configuration with points at u ± ε ⁢ v and - u ± ε ⁢ w . We get a single transversal intersection when u, v and w are all the equivalence class of ( 0 , 0 , 1 ) . To evaluate [ 2 , 1 , 2 , … , 1 , 2 ] on q 1 , … , 1 , 2 , we represent the latter by a standard map of the iterated wreath product S 1 ⁢ ∫ ⋯ ⁢ ∫ ℝ ⁢ P 2 , where

to Conf ¯ 2 ℓ ⁢ ( ℝ ∞ ) . Once again, we will get a single transversal intersection when each coordinate entry of S 1 is ( 0 , ± 1 ) and each of ℝ ⁢ P 2 is the equivalence class of ( 0 , 0 , 1 ) .

To pass to higher m, we perform an induction based on the detection result of Madsen and Milgram [18], that the cohomology of symmetric groups is detected by coproduct, along with restriction to V n for 𝒮 2 n . The formulas for coproducts of γ 2 , m 2 and α 2 , m ⁢ ( 2 ) (respectively, γ ℓ , m ⋅ γ 1 , m ⁢ 2 ℓ - 1 and α ℓ , m ⁢ ( 1.5 ) ), the former given by Theorem 3.1 and Hopf semi-ring distributivity and the latter given by Theorem 4.16, will be equal by inductive assumption, and application of detection establishes the induction step when m ≠ 2 n . When m = 2 n , we note that by having the same coproducts they must agree up to the kernel of restriction to V n , which is generated by γ n , 1 . But there are no non-zero multiples of γ n , 1 in the relevant degrees. ∎