2017-06-09 21:48:00 +00:00
|
|
|
|
/-
|
|
|
|
|
|
Copyright (c) 2017 Kuen-Bang Hou (Favonia).
|
|
|
|
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
|
|
|
|
|
|
|
|
Author: Kuen-Bang Hou (Favonia)
|
|
|
|
|
|
-/
|
|
|
|
|
|
|
2017-07-17 12:58:36 +00:00
|
|
|
|
import .basic .sphere ..homotopy.susp_product
|
2017-06-09 21:48:00 +00:00
|
|
|
|
|
|
|
|
|
|
open eq pointed group algebra circle sphere nat equiv susp
|
|
|
|
|
|
function sphere homology int lift prod smash
|
|
|
|
|
|
|
|
|
|
|
|
namespace homology
|
|
|
|
|
|
|
|
|
|
|
|
section
|
|
|
|
|
|
parameter (theory : ordinary_homology_theory)
|
|
|
|
|
|
|
|
|
|
|
|
open ordinary_homology_theory
|
|
|
|
|
|
|
2017-07-20 17:01:22 +00:00
|
|
|
|
theorem Hptorus : Π(n : ℤ)(m : ℕ), HH theory n (plift (sphere m ×* sphere m)) ≃g
|
|
|
|
|
|
HH theory n (plift (sphere m)) ×g (HH theory n (plift (sphere m)) ×g HH theory n (plift (sphere (m + m)))) := λ n m,
|
|
|
|
|
|
calc HH theory n (plift (sphere m ×* sphere m))
|
|
|
|
|
|
≃g HH theory (n + 1) (plift (⅀ (sphere m ×* sphere m))) : by exact (Hplift_susp theory n (sphere m ×* sphere m))⁻¹ᵍ
|
|
|
|
|
|
... ≃g HH theory (n + 1) (plift (⅀ (sphere m) ∨ (⅀ (sphere m) ∨ ⅀ (sphere m ∧ sphere m))))
|
|
|
|
|
|
: by exact Hplift_isomorphism theory (n + 1) (susp_product (sphere m) (sphere m))
|
|
|
|
|
|
... ≃g HH theory (n + 1) (plift (⅀ (sphere m))) ×g HH theory (n + 1) (plift (⅀ (sphere m) ∨ (⅀ (sphere m ∧ sphere m))))
|
|
|
|
|
|
: by exact Hplift_wedge theory (n + 1) (⅀ (sphere m)) (⅀ (sphere m) ∨ (⅀ (sphere m ∧ sphere m)))
|
|
|
|
|
|
... ≃g HH theory n (plift (sphere m)) ×g (HH theory n (plift (sphere m)) ×g HH theory n (plift (sphere (m + m))))
|
|
|
|
|
|
: by exact product_isomorphism (Hplift_susp theory n (sphere m))
|
2017-06-09 21:48:00 +00:00
|
|
|
|
(calc
|
2017-07-20 17:01:22 +00:00
|
|
|
|
HH theory (n + 1) (plift (⅀ (sphere m) ∨ (⅀ (sphere m ∧ sphere m))))
|
|
|
|
|
|
≃g HH theory (n + 1) (plift (⅀ (sphere m))) ×g HH theory (n + 1) (plift (⅀ (sphere m ∧ sphere m)))
|
|
|
|
|
|
: by exact Hplift_wedge theory (n + 1) (⅀ (sphere m)) (⅀ (sphere m ∧ sphere m))
|
|
|
|
|
|
... ≃g HH theory n (plift (sphere m)) ×g HH theory n (plift (sphere (m + m)))
|
|
|
|
|
|
: by exact product_isomorphism (Hplift_susp theory n (sphere m))
|
|
|
|
|
|
(Hplift_susp theory n (sphere m ∧ sphere m) ⬝g Hplift_isomorphism theory n (sphere_smash_sphere m m)))
|
2017-06-09 21:48:00 +00:00
|
|
|
|
|
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
|
|
end homology
|
