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Add Hptorus.

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favonia 2017-06-09 15:48:00 -06:00
parent b8de7ffd80
commit a78c92636e
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18
homology/homology.hlean
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@ -6,10 +6,10 @@ Authors: Yuri Sulyma, Favonia
Reduced homology theories
-/
import ..homotopy.spectrum ..homotopy.EM ..algebra.arrow_group ..algebra.direct_sum ..homotopy.fwedge ..choice ..homotopy.pushout ..move_to_lib
import ..homotopy.spectrum ..homotopy.EM ..algebra.arrow_group ..algebra.direct_sum ..homotopy.fwedge ..homotopy.wedge ..choice ..homotopy.pushout ..move_to_lib
open eq spectrum int pointed group algebra sphere nat equiv susp is_trunc
function fwedge cofiber lift is_equiv choice algebra pi smash
function fwedge cofiber lift is_equiv choice algebra pi smash wedge
namespace homology
@ -129,6 +129,20 @@ namespace homology
... ≃g HH theory n A ×g HH theory n B : by exact binary_dirsum (HH theory n A) (HH theory n B)
end
section
universe variables u v
parameter (theory : homology_theory.{max u v})
open homology_theory
definition Hplift_psusp (n : ℤ) (A : Type*): HH theory (n + 1) (plift.{u v} (psusp A)) ≃g HH theory n (plift.{u v} A) :=
calc HH theory (n + 1) (plift.{u v} (psusp A)) ≃g HH theory (n + 1) (psusp (plift.{u v} A)) : by exact HH_isomorphism theory (n + 1) (plift_psusp _)
... ≃g HH theory n (plift.{u v} A) : by exact Hpsusp theory n (plift.{u v} A)
definition Hplift_pwedge (n : ℤ) (A B : Type*): HH theory n (plift.{u v} (A ∨ B)) ≃g HH theory n (plift.{u v} A) ×g HH theory n (plift.{u v} B) :=
calc HH theory n (plift.{u v} (A ∨ B)) ≃g HH theory n (plift.{u v} A ∨ plift.{u v} B) : by exact HH_isomorphism theory n (plift_pwedge A B)
... ≃g HH theory n (plift.{u v} A) ×g HH theory n (plift.{u v} B) : by exact Hpwedge theory n (plift.{u v} A) (plift.{u v} B)
end
/- homology theory associated to a prespectrum -/
definition homology (X : Type*) (E : prespectrum) (n : ℤ) : AbGroup :=

7
homology/sphere.hlean
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@ -23,10 +23,9 @@ section
{ 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 Hpsusp theory (pred n) (plift (psphere m))
≃g HH theory (succ (pred n)) (plift (psusp (psphere m)))
: by exact isomorphism_ap (λ n, HH theory n (plift (psusp (psphere m)))) (succ_pred n)⁻¹
... ≃g HH theory (pred n) (plift (psphere m)) : by exact Hplift_psusp theory (pred n) (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))

40
homology/torus.hlean Normal file
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@ -0,0 +1,40 @@
/-
Copyright (c) 2017 Kuen-Bang Hou (Favonia).
Released under Apache 2.0 license as described in the file LICENSE.
Author: Kuen-Bang Hou (Favonia)
-/
import .homology .sphere ..susp_product
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
theorem Hptorus : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m ×* psphere m)) ≃g
HH theory n (plift (psphere m)) ×g (HH theory n (plift (psphere m)) ×g HH theory n (plift (psphere (m + m)))) := λ n m,
calc HH theory n (plift (psphere m ×* psphere m))
≃g HH theory (n + 1) (plift (⅀ (psphere m ×* psphere m))) : by exact (Hplift_psusp theory n (psphere m ×* psphere m))⁻¹ᵍ
... ≃g HH theory (n + 1) (plift (⅀ (psphere m) ∨ (⅀ (psphere m) ∨ ⅀ (psphere m ∧ psphere m))))
: by exact HH_isomorphism theory (n + 1) (!pequiv_plift⁻¹ᵉ* ⬝e* susp_product (psphere m) (psphere m) ⬝e* !pequiv_plift)
... ≃g HH theory (n + 1) (plift (⅀ (psphere m))) ×g HH theory (n + 1) (plift (⅀ (psphere m) ∨ (⅀ (psphere m ∧ psphere m))))
: by exact Hplift_pwedge theory (n + 1) (⅀ (psphere m)) (⅀ (psphere m) ∨ (⅀ (psphere m ∧ psphere m)))
... ≃g HH theory n (plift (psphere m)) ×g (HH theory n (plift (psphere m)) ×g HH theory n (plift (psphere (m + m))))
: by exact product_isomorphism (Hplift_psusp theory n (psphere m))
(calc
HH theory (n + 1) (plift (⅀ (psphere m) ∨ (⅀ (psphere m ∧ psphere m))))
≃g HH theory (n + 1) (plift (⅀ (psphere m))) ×g HH theory (n + 1) (plift (⅀ (psphere m ∧ psphere m)))
: by exact Hplift_pwedge theory (n + 1) (⅀ (psphere m)) (⅀ (psphere m ∧ psphere m))
... ≃g HH theory n (plift (psphere m)) ×g HH theory n (plift (psphere (m + m)))
: by exact product_isomorphism (Hplift_psusp theory n (psphere m))
(Hplift_psusp theory n (psphere m ∧ psphere m) ⬝g HH_isomorphism theory n (!pequiv_plift⁻¹ᵉ* ⬝e* (sphere_smash_sphere m m) ⬝e* !pequiv_plift)))
end
end homology