37 lines
1.4 KiB
Text
37 lines
1.4 KiB
Text
/-
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Copyright (c) 2017 Kuen-Bang Hou (Favonia).
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Kuen-Bang Hou (Favonia)
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-/
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import .homology
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open eq pointed group algebra circle sphere nat equiv susp
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function sphere homology int lift
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namespace homology
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section
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parameter (theory : homology_theory)
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open homology_theory
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theorem Hpsphere : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m)) ≃g HH theory (n - m) (plift (psphere 0)) :=
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begin
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intros n m, revert n, induction m with m,
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{ exact λ n, isomorphism_ap (λ n, HH theory n (plift (psphere 0))) (sub_zero n)⁻¹ },
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{ intro n, exact calc
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HH theory n (plift (psusp (psphere m)))
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≃g HH theory n (psusp (plift (psphere m))) : by exact HH_isomorphism theory n (plift_psusp (psphere m))
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... ≃g HH theory (succ (pred n)) (psusp (plift (psphere m)))
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: by exact isomorphism_ap (λ n, HH theory n (psusp (plift (psphere m)))) (succ_pred n)⁻¹
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... ≃g HH theory (pred n) (plift (psphere m)) : by exact Hpsusp theory (pred n) (plift (psphere m))
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... ≃g HH theory (pred n - m) (plift (psphere 0)) : by exact v_0 (pred n)
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... ≃g HH theory (n - succ m) (plift (psphere 0))
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: 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))
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}
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end
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end
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end homology
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