Matching Tag: multiple-covers
Let S S S β ∈ H 2 ( S , Z ) \beta\in H_2(S,\mathbb{Z}) β ∈ H 2 ( S , Z ) ⟨ τ k 1 ( γ 1 ) ⋯ τ k n ( γ n ) ⟩ g , β S \langle \tau_{k_1}(\gamma_1) \cdots \tau_{k_n}(\gamma_n) \rangle^S_{g,\beta} ⟨ τ k 1 ( γ 1 ) ⋯ τ k n ( γ n ) ⟩ g , β S
Let A A A β ∈ H 2 ( A , Z ) \beta\in H_2(A,\mathbb{Z}) β ∈ H 2 ( A , Z ) k ∣ β k\mid\beta k ∣ β A k A_k A k
Let X X X K 3 [ n ] K3^{[n]} K 3 [ n ] β ∈ H 2 ( X , Z ) \beta\in H_2(X,\mathbb{Z}) β ∈ H 2 ( X , Z ) k ∣ β k\mid\beta k ∣ β X k X_k X k K 3 [ n ] K3^{[n]} K 3 [ n ]
Let S S S α \alpha α χ S ( α , α ) \chi_S(\alpha,\alpha) χ S ( α , α ) [ r ] t = ( t r / 2 − t − r / 2 ) / ( t 1 / 2 − t − 1 / 2 ) [r]_t=(t^{r/2}-t^{-r/2})/(t^{1/2}-t^{-1/2}) [ r ] t = ( t r /2 − t − r /2 ) / ( t 1/2 − t − 1/2 )
Multiple-cover formula. For all g ≥ 2 g\geq2 g ≥ 2 d 1 , d 2 > 0 d_1,d_2>0 d 1 , d 2 > 0 d 3 ≥ 0 d_3\geq0 d 3 ≥ 0
Let S S S Γ 0 \Gamma_0 Γ 0 ( v , v ) (v,v) ( v , v ) k ∣ v k\mid v k ∣ v
Let S S S H ~ ( S , Z ) = Z ⊕ H 2 ( S , Z ) ⊕ Z \widetilde{H}(S,\mathbb{Z})=\mathbb{Z}\oplus H^2(S,\mathbb{Z})\oplus\mathbb{Z} H ( S , Z ) = Z ⊕ H 2 ( S , Z ) ⊕ Z v v v
