Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Long exact sequence of homotopy groups + Generator of π₃S² #698

Merged
merged 173 commits into from
Jan 27, 2022

Conversation

aljungstrom
Copy link
Contributor

This PR contains

  1. The long exact sequence of homotopy groups (and corresponding loop spaces). It also contains some information about how the maps in the sequence look when they are transported to the (Sⁿ→∙A) definition of homotopy groups. Right now exactness is stated in the usual way by having a bunch of separate theorems on the form Ker⊂Im and Im⊂Ker, like in Mayer-Vietrois and Gysin. Anders and I have been discussing adding a reasonable definition of exact sequences for some time now. When that's a reality, things will look nicer.
  2. A proof of π₃S²≅ℤ together with the fact that the Hopf map is the generator of π₃S².
  3. A proof of πₙSⁿ≅ℤ together with the fact that the identity map is the generator of πₙSⁿ (needed above). The construction of the Iso is somewhat roundabout to make the characterisation of the generator easier. The characterisation of the generator is longish for π₂S² with, but easy in all other cases. Hopefully in the future this step will hold simply by computation.

Cubical/Homotopy/Group/Base.agda Show resolved Hide resolved
Cubical/Homotopy/Group/LES.agda Outdated Show resolved Hide resolved
Cubical/Homotopy/Group/LES.agda Outdated Show resolved Hide resolved
Cubical/Homotopy/Group/LES.agda Outdated Show resolved Hide resolved
Cubical/Homotopy/Group/LES.agda Outdated Show resolved Hide resolved
Cubical/Homotopy/Group/Pi4S3/Summary.agda Outdated Show resolved Hide resolved
Cubical/Homotopy/Group/PinSn.agda Outdated Show resolved Hide resolved
Cubical/Homotopy/Group/PinSn.agda Outdated Show resolved Hide resolved
Cubical/Homotopy/Group/PinSn.agda Outdated Show resolved Hide resolved
Cubical/Homotopy/Group/PinSn.agda Show resolved Hide resolved
@aljungstrom
Copy link
Contributor Author

@mortberg Ready!

@mortberg mortberg merged commit 9acdecf into agda:master Jan 27, 2022
Sign up for free to join this conversation on GitHub. Already have an account? Sign in to comment
Labels
None yet
Projects
None yet
Development

Successfully merging this pull request may close these issues.

None yet

2 participants