93 lines
4.2 KiB
Markdown
93 lines
4.2 KiB
Markdown
# Spectral Sequences
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Formalization project of the CMU HoTT group towards formalizing the Serre spectral sequence.
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#### Participants
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Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Floris van Doorn, Clive Newstead, Egbert Rijke, Mike Shulman.
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## Resources
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- [Mike's blog posts on ncatlab](https://ncatlab.org/homotopytypetheory/show/spectral+sequences).
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- The [Licata-Finster article](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf) about Eilenberg-Mac Lane spaces.
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- We learned about the Serre spectral sequence from [Hatcher's chapter about spectral sequences](https://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html).
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- 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.
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- Mac Lane's Homology contains a lot of homological algebra and a chapter on spectral sequences, including exact couples.
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## Contents for Lean spectral sequences project
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### Outline
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These projects are mostly done
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- Given a sequence of spectra and maps, indexed over `ℤ`, we get an exact couple, indexed over `ℤ × ℤ`.
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- We can derive an exact couple.
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- If the exact couple is bounded, we repeat this process to get a convergent spectral sequence.
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- We construct the Atiyah-Hirzebruch and Serre spectral sequences for cohomology.
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### Future directions
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- Hurewicz Theorem and Hurewicz theorem modulo a Serre class.
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- Homological Serre spectral sequence.
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- Interaction between steenrod squares and cup product with spectral sequences
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- ...
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### Algebra
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#### To do
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- Constructions: tensor, hom, projective, Tor (at least on groups)
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- Finite groups, Finitely generated groups, torsion groups
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- Serre classes
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- [vector spaces](http://ncatlab.org/nlab/show/vector+space),
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#### In Progress
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#### Done
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- groups, rings, fields, [R-modules](http://ncatlab.org/nlab/show/module), graded R-modules.
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- Constructions on groups and abelian groups:: subgroup, quotient, product, free groups.
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- Constructions on ablian groups: direct sum, sequential colimi.
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- exact sequences, short and long.
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- [chain complexes](http://ncatlab.org/nlab/show/chain+complex) and [homology](http://ncatlab.org/nlab/show/homology).
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- [exact couples](http://ncatlab.org/nlab/show/exact+couple) graded over an arbitrary indexing set.
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- spectral sequence of an exact couple.
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- [convergence of spectral sequences](http://ncatlab.org/nlab/show/spectral+sequence#ConvergenceOfSpectralSequences).
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### Topology
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#### To do
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- cofiber sequences
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+ Hom'ing out gives a fiber sequence: if `A → B → coker f` cofiber
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sequences, then `X^A → X^B → X^(coker f)` is a fiber sequence.
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- fiber and cofiber sequences of spectra, stability
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+ limits are levelwise
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+ colimits need to be spectrified
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- long exact sequence from cofiber sequences of spectra
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+ indexed on ℤ, need to splice together LES's
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- Cup product on cohomology groups
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- Parametrized and unreduced homology
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- Steenrod squares
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- ...
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#### In Progress
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- [prespectra](http://ncatlab.org/nlab/show/spectrum+object) and [spectra](http://ncatlab.org/nlab/show/spectrum), indexed over an arbitrary type with a successor
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+ think about equivariant spectra indexed by representations of `G`
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- [spectrification](http://ncatlab.org/nlab/show/higher+inductive+type#spectrification)
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+ adjoint to forgetful
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+ as sequential colimit, prove induction principle
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+ connective spectrum: `is_conn n.-2 Eₙ`
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- Postnikov towers of spectra.
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+ basic definition already there
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+ fibers of Postnikov sequence unstably and stably
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- [parametrized spectra](http://ncatlab.org/nlab/show/parametrized+spectrum), parametrized smash and hom between types and spectra.
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- Check Eilenberg-Steenrod axioms for reduced homology.
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#### Done
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- Most things in the HoTT Book up to Section 8.9 (see [this file](https://github.com/leanprover/lean/blob/master/hott/book.md))
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- pointed types, maps, homotopies and equivalences
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- [Eilenberg-MacLane spaces](http://ncatlab.org/nlab/show/Eilenberg-Mac+Lane+space) and EM-spectrum
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- fiber sequence
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+ already have the LES
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+ need shift isomorphism
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+ Hom'ing into a fiber sequence gives another fiber sequence.
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- long exact sequence of homotopy groups of spectra, indexed on ℤ
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- exact couple of a tower of spectra
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+ need to splice together LES's
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