Cohomology ring of a moment-angle complex #36227
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Documentation preview for this PR (built with commit ffb12b6; changes) is ready! 🎉 |
tscrim
requested changes
Feb 3, 2025
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| def one(self): | ||
| """ |
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Suggested change
| def one(self): | |
| """ | |
| @cached_method | |
| def one(self): | |
| """ |
So we don't have to repeatedly recompute it as it can be called often.
Actually, this is always just a single element one_basis() and return that index (0, 0). This seems to be an assumption of _to_cycle_on_basis and product_on_basis.
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| if self.is_one(): | ||
| return self.parent()._complex.simplicial_complex().generated_subcomplex(set(), is_mutable=is_mutable) | ||
| vertex_set = self.parent()._gens[self.leading_support()[0]][self.leading_support()[1]][0] | ||
| return self.parent()._complex.simplicial_complex().generated_subcomplex(vertex_set, is_mutable=is_mutable) |
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| if self.is_one(): | |
| return self.parent()._complex.simplicial_complex().generated_subcomplex(set(), is_mutable=is_mutable) | |
| vertex_set = self.parent()._gens[self.leading_support()[0]][self.leading_support()[1]][0] | |
| return self.parent()._complex.simplicial_complex().generated_subcomplex(vertex_set, is_mutable=is_mutable) | |
| P = self.parent() | |
| if self.is_one(): | |
| return P._complex.simplicial_complex().generated_subcomplex(set(), is_mutable=is_mutable) | |
| ls = self.leading_support() | |
| vertex_set = P._gens[ls[0]][ls[1]][0] | |
| return P._complex.simplicial_complex().generated_subcomplex(vertex_set, is_mutable=is_mutable) |
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| res = 1 | ||
| for element in subcomplex.vertices(): | ||
| res *= _eps(element, simplicial_complex) | ||
| return res |
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I don't see the need for the _eps method. I would just put it inline here. This can also be simplified
Suggested change
| res = 1 | |
| for element in subcomplex.vertices(): | |
| res *= _eps(element, simplicial_complex) | |
| return res | |
| return (-1) ** sum(simplicial_complex._vertex_to_index[element] | |
| for element in subcomplex.vertices()) |
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| return res | ||
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| def cup_length(self): |
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| def cup_length(self): | |
| @cached_method | |
| def cup_length(self, elements=False): |
It would be good to have an option to return the list of elements that give the cup length.
| cochains_basis = subcomplex_union.n_chains(deg, cochains=True).basis() | ||
| # If the resulting cochain exists, then we | ||
| # add it to the result; otherwise, we add 0 | ||
| if cochains_basis.has_key(Simplex(res_union)): |
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| if cochains_basis.has_key(Simplex(res_union)): | |
| X = Simplex(res_union) | |
| if X in cochains_basis: |
since cochains_basis is a dict, right? This way you don't have to recreate the Simplex below either. Although I would do this as
if X not in cochains_basis:
continue
[rest of code]
Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>
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This PR provides the
MomentAngleComplexclass with its own cohomology ring (similar to the one which is available for simplicial, cubical and delta complexes). It requires its own separate class because we do not wish to compute this space explicitly (for efficiency reasons), but rather use a certain isomorphism which allows us to compute the cohomology ring more efficiently. The added class (CohomologyRing) is designed to mimic the already implemented class for cohomology rings by adding as many analogue methods as possible (most significant of which allows us to compute the cup product), but its structure differs fundamentally, and therefore must be a separate class.This is a part of #35640 (GSoC 2023).
📝 Checklist
⌛ Dependencies