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Add Hsphere.

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favonia 2017-06-06 17:30:42 -06:00
parent a292dba89f
commit bf8f77a9e5
1 changed files with 17 additions and 5 deletions
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homology/sphere.hlean
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@ -1,7 +1,7 @@
-- Author: Kuen-Bang Hou (Favonia) -- Author: Kuen-Bang Hou (Favonia)
import core types.lift import core
import ..homotopy.homology import .homology
open eq pointed group algebra circle sphere nat equiv susp open eq pointed group algebra circle sphere nat equiv susp
function sphere homology int lift function sphere homology int lift
@ -13,9 +13,21 @@ section
open homology_theory open homology_theory
theorem Hsphere : Π(n : ℕ), HH theory n (plift (psphere n)) ≃g HH theory 0 (plift (psphere 0)) := theorem Hsphere : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m)) ≃g HH theory (n - m) (plift (psphere 0)) :=
sorry 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
end homology end homology