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cohomology of simplicial complexes with finite coefficients as modules over steenrod algebras #6103

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antieau mannequin opened this issue May 21, 2009 · 1 comment · Fixed by #36310
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cohomology of simplicial complexes with finite coefficients as modules over steenrod algebras #6103

antieau mannequin opened this issue May 21, 2009 · 1 comment · Fixed by #36310
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@jhpalmieri
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c: algebraic topology p: minor / 4 t: enhancement
Milestone
sage-10.2

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@antieau
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antieau mannequin commented May 21, 2009

Implement functionality in sage to return the ZZ-graded cohomology with GF(p)-coefficients of an arbitrary simplicial complex as a module over the Steenrod algebra A(p).

This will depend on #6099, #6100, #6101, and #5882. The methodolgy will be similar to that implemented in the ticket #6102.

CC: @jhpalmieri @antieau

Component: algebraic topology

Issue created by migration from https://trac.sagemath.org/ticket/6103
@antieau antieau mannequin assigned jhpalmieri May 21, 2009
@jhpalmieri jhpalmieri mentioned this issue Apr 7, 2011
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@jhpalmieri jhpalmieri mentioned this issue Sep 21, 2023
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Note that #36310 only does that at the prime 2, not in general. Still good enough (IMHO) to close this, if #36310 is merged.

vbraun pushed a commit to vbraun/sage that referenced this issue Oct 1, 2023
sagemathgh-36310: Mod 2 (co)homology as a module over the Steenrod al…
274c55e 
…gebra

    
Mod 2 (co)homology as a module over the Steenrod algebra

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Implement:

- vector space duality for (co)homology of cell complexes with field
coefficients, switching between homology and cohomology
- Steenrod operations on homology and cohomology, acting from the left
and from the right

The Steenrod operation "Sq(i)" was already implemented for cohomology
rings, raising an error if it was used with a base field other than
GF(2). Now there is a new class, `CohomologyRing_mod2`, which should
only be used with that base field, and the `Sq` method is attached to
Elements of that new class.

Using Steenrod operations is more transparent: in addition to doing
`x.Sq(i)`, users can do `Sq(i) * x` or `x * Sq(i)`, or indeed `a * x` or
`x * a` for any element `a` of `SteenrodAlgebra(2)`. (The elements
`Sq(i)` generate the Steenrod algebra, but they do not constitute a full
list of the elements.)

This fixes sagemath#6103.

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### ⌛ Dependencies

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URL: sagemath#36310
Reported by: John H. Palmieri
Reviewer(s): John H. Palmieri, Travis Scrimshaw
vbraun pushed a commit that referenced this issue Oct 8, 2023
gh-36310: Mod 2 (co)homology as a module over the Steenrod algebra
fe963ca 
    
Mod 2 (co)homology as a module over the Steenrod algebra

<!-- ^^^^^
Please provide a concise, informative and self-explanatory title.
Don't put issue numbers in there, do this in the PR body below.
For example, instead of "Fixes #1234" use "Introduce new method to
calculate 1+1"
-->
<!-- Describe your changes here in detail -->

Implement:

- vector space duality for (co)homology of cell complexes with field
coefficients, switching between homology and cohomology
- Steenrod operations on homology and cohomology, acting from the left
and from the right

The Steenrod operation "Sq(i)" was already implemented for cohomology
rings, raising an error if it was used with a base field other than
GF(2). Now there is a new class, `CohomologyRing_mod2`, which should
only be used with that base field, and the `Sq` method is attached to
Elements of that new class.

Using Steenrod operations is more transparent: in addition to doing
`x.Sq(i)`, users can do `Sq(i) * x` or `x * Sq(i)`, or indeed `a * x` or
`x * a` for any element `a` of `SteenrodAlgebra(2)`. (The elements
`Sq(i)` generate the Steenrod algebra, but they do not constitute a full
list of the elements.)

This fixes #6103.

<!-- Why is this change required? What problem does it solve? -->
<!-- If this PR resolves an open issue, please link to it here. For
example "Fixes #12345". -->
<!-- If your change requires a documentation PR, please link it
appropriately. -->

### 📝 Checklist

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appropriately -->
<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
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- [X] The title is concise, informative, and self-explanatory.
- [X] The description explains in detail what this PR is about.
- [X] I have linked a relevant issue or discussion.
- [X] I have created tests covering the changes.
- [X] I have updated the documentation accordingly.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on
- #12345: short description why this is a dependency
- #34567: ...
-->

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URL: #36310
Reported by: John H. Palmieri
Reviewer(s): John H. Palmieri, Travis Scrimshaw
@mkoeppe mkoeppe added this to the sage-10.2 milestone Oct 8, 2023
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Mod 2 (co)homology as a module over the Steenrod algebra jhpalmieri/sage
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