101 lines
5 KiB
Markdown
101 lines
5 KiB
Markdown
# Spectral Sequences
|
||
|
||
Formalization project of the CMU HoTT group towards formalizing the Serre spectral sequence.
|
||
|
||
This repository also contains the contents of the MRC group on formalizing homology in Lean.
|
||
|
||
#### Participants
|
||
Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Floris van Doorn, Clive Newstead, Egbert Rijke, Mike Shulman.
|
||
|
||
## Resources
|
||
- [Mike's blog posts on ncatlab](https://ncatlab.org/homotopytypetheory/show/spectral+sequences).
|
||
- The [Licata-Finster article](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf) about Eilenberg-Mac Lane spaces.
|
||
- We learned about the Serre spectral sequence from [Hatcher's chapter about spectral sequences](https://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html).
|
||
- Lang's algebra (revised 3rd edition) contains a chapter on general homology theory, with a section on spectral sequences. Thus, we can use this book at least as an outline for the algebraic part of the project.
|
||
- Mac Lane's Homology contains a lot of homological algebra and a chapter on spectral sequences, including exact couples.
|
||
|
||
## Contents for Lean spectral sequences project
|
||
|
||
### Outline
|
||
|
||
These projects are mostly done
|
||
|
||
- Given a sequence of spectra and maps, indexed over `ℤ`, we get an exact couple, indexed over `ℤ × ℤ`.
|
||
- We can derive an exact couple.
|
||
- If the exact couple is bounded, we repeat this process to get a convergent spectral sequence.
|
||
- We construct the Atiyah-Hirzebruch and Serre spectral sequences for cohomology.
|
||
|
||
### Future directions
|
||
- Hurewicz Theorem and Hurewicz theorem modulo a Serre class. There is a proof in Hatcher. Also, [this](http://www.math.uni-frankfurt.de/~johannso/SkriptAll/SkriptTopAlg/SkriptTopCW/homotop12.pdf) might be useful.
|
||
- Homological Serre spectral sequence.
|
||
- Interaction between steenrod squares and cup product with spectral sequences
|
||
- ...
|
||
|
||
### Algebra
|
||
|
||
#### To do
|
||
- Constructions: tensor, hom, projective, Tor (at least on groups)
|
||
- Finite groups, Finitely generated groups, torsion groups
|
||
- Serre classes
|
||
- [vector spaces](http://ncatlab.org/nlab/show/vector+space),
|
||
|
||
#### In Progress
|
||
|
||
|
||
#### Done
|
||
- groups, rings, fields, [R-modules](http://ncatlab.org/nlab/show/module), graded R-modules.
|
||
- Constructions on groups and abelian groups:: subgroup, quotient, product, free groups.
|
||
- Constructions on ablian groups: direct sum, sequential colimi.
|
||
- exact sequences, short and long.
|
||
- [chain complexes](http://ncatlab.org/nlab/show/chain+complex) and [homology](http://ncatlab.org/nlab/show/homology).
|
||
- [exact couples](http://ncatlab.org/nlab/show/exact+couple) graded over an arbitrary indexing set.
|
||
- spectral sequence of an exact couple.
|
||
- [convergence of spectral sequences](http://ncatlab.org/nlab/show/spectral+sequence#ConvergenceOfSpectralSequences).
|
||
|
||
### Topology
|
||
|
||
#### To do
|
||
- cofiber sequences
|
||
+ Hom'ing out gives a fiber sequence: if `A → B → coker f` cofiber
|
||
sequences, then `X^A → X^B → X^(coker f)` is a fiber sequence.
|
||
- fiber and cofiber sequences of spectra, stability
|
||
+ limits are levelwise
|
||
+ colimits need to be spectrified
|
||
- long exact sequence from cofiber sequences of spectra
|
||
+ indexed on ℤ, need to splice together LES's
|
||
- Cup product on cohomology groups
|
||
- Parametrized and unreduced homology
|
||
- Steenrod squares
|
||
- ...
|
||
|
||
#### In Progress
|
||
- [prespectra](http://ncatlab.org/nlab/show/spectrum+object) and [spectra](http://ncatlab.org/nlab/show/spectrum), indexed over an arbitrary type with a successor
|
||
+ think about equivariant spectra indexed by representations of `G`
|
||
- [spectrification](http://ncatlab.org/nlab/show/higher+inductive+type#spectrification)
|
||
+ adjoint to forgetful
|
||
+ as sequential colimit, prove induction principle
|
||
+ connective spectrum: `is_conn n.-2 Eₙ`
|
||
- Postnikov towers of spectra.
|
||
+ basic definition already there
|
||
+ fibers of Postnikov sequence unstably and stably
|
||
- [parametrized spectra](http://ncatlab.org/nlab/show/parametrized+spectrum), parametrized smash and hom between types and spectra.
|
||
- Check Eilenberg-Steenrod axioms for reduced homology.
|
||
|
||
|
||
#### Done
|
||
- Most things in the HoTT Book up to Section 8.9 (see [this file](https://github.com/leanprover/lean/blob/master/hott/book.md))
|
||
- pointed types, maps, homotopies and equivalences
|
||
- [Eilenberg-MacLane spaces](http://ncatlab.org/nlab/show/Eilenberg-Mac+Lane+space) and EM-spectrum
|
||
- fiber sequence
|
||
+ already have the LES
|
||
+ need shift isomorphism
|
||
+ Hom'ing into a fiber sequence gives another fiber sequence.
|
||
- long exact sequence of homotopy groups of spectra, indexed on ℤ
|
||
- exact couple of a tower of spectra
|
||
+ need to splice together LES's
|
||
|
||
## Contributing
|
||
- We will try to make sure that this repository compiles with the newest version of Lean 2.
|
||
- Installation instructions for Lean 2 can be found [here](https://github.com/leanprover/lean2).
|
||
- Some notes on the Emacs mode can be found [here](https://github.com/leanprover/lean2/blob/master/src/emacs/README.md) (for example if some unicode characters don't show up, or increase the spacing between lines by a lot).
|
||
- If you contribute, please use rebase instead of merge (e.g. `git pull -r`).
|