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Add Hsphere.

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favonia 2017-06-06 17:30:42 -06:00
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homology/sphere.hlean
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@ -1,7 +1,7 @@
-- Author: Kuen-Bang Hou (Favonia)
import core types.lift
import ..homotopy.homology
import core
import .homology
open eq pointed group algebra circle sphere nat equiv susp
function sphere homology int lift
@ -13,9 +13,21 @@ section
open homology_theory
theorem Hsphere : Π(n : ℕ), HH theory n (plift (psphere n)) ≃g HH theory 0 (plift (psphere 0)) :=
sorry
theorem Hsphere : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m)) ≃g HH theory (n - m) (plift (psphere 0)) :=
begin
intros n m, revert n, induction m with m,
{ exact λ n, isomorphism_ap (λ n, HH theory n (plift (psphere 0))) (sub_zero n)⁻¹ },
{ intro n, exact calc
HH theory n (plift (psusp (psphere m)))
≃g HH theory n (psusp (plift (psphere m))) : by exact HH_isomorphism theory n (plift_psusp (psphere m))
... ≃g HH theory (succ (pred n)) (psusp (plift (psphere m)))
: by exact isomorphism_ap (λ n, HH theory n (psusp (plift (psphere m)))) (succ_pred n)⁻¹
... ≃g HH theory (pred n) (plift (psphere m)) : by exact Hsusp theory (pred n) (plift (psphere m))
... ≃g HH theory (pred n - m) (plift (psphere 0)) : by exact v_0 (pred n)
... ≃g HH theory (n - succ m) (plift (psphere 0))
: by exact isomorphism_ap (λ n, HH theory n (plift (psphere 0))) (sub_sub n 1 m ⬝ ap (λ m, n - m) (add.comm 1 m))
}
end
end
end homology