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src/sage/topology: remove all "needs sage.foo" tags #42071

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1 change: 0 additions & 1 deletion src/sage/topology/all.py
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Original file line number Diff line number Diff line change
@@ -1,4 +1,3 @@
# sage.doctest: needs sage.graphs
from sage.topology.simplicial_complex import SimplicialComplex, Simplex

from sage.topology.simplicial_complex_morphism import SimplicialComplexMorphism
Expand Down
68 changes: 30 additions & 38 deletions src/sage/topology/cell_complex.py
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Original file line number Diff line number Diff line change
@@ -1,4 +1,3 @@
# sage.doctest: needs sage.graphs
r"""
Generic cell complexes

Expand Down Expand Up @@ -228,7 +227,6 @@ def _n_cells_sorted(self, n, subcomplex=None):
sage: Z._n_cells_sorted(2, subcomplex=K)
[(1, 2, 4), (1, 3, 4)]

sage: # needs sage.symbolic
sage: S = SimplicialComplex([[complex(i), complex(1)]])
sage: S._n_cells_sorted(0)
[((1+0j),), (1j,)]
Expand Down Expand Up @@ -508,7 +506,6 @@ def homology(self, dim=None, base_ring=ZZ, subcomplex=None,

EXAMPLES::

sage: # needs sage.modules
sage: P = delta_complexes.RealProjectivePlane()
sage: P.homology()
{0: 0, 1: C2, 2: 0}
Expand All @@ -527,7 +524,7 @@ def homology(self, dim=None, base_ring=ZZ, subcomplex=None,
Sage can compute generators of homology groups::

sage: S2 = simplicial_complexes.Sphere(2)
sage: S2.homology(dim=2, generators=True, base_ring=GF(2)) # needs sage.modules
sage: S2.homology(dim=2, generators=True, base_ring=GF(2))
[(Vector space of dimension 1 over Finite Field of size 2,
(0, 1, 2) + (0, 1, 3) + (0, 2, 3) + (1, 2, 3))]

Expand All @@ -538,15 +535,15 @@ def homology(self, dim=None, base_ring=ZZ, subcomplex=None,
complexes, each generator is a linear combination of cubes::

sage: S2_cub = cubical_complexes.Sphere(2)
sage: S2_cub.homology(dim=2, generators=True) # needs sage.modules
sage: S2_cub.homology(dim=2, generators=True)
[(Z,
[0,0] x [0,1] x [0,1] - [0,1] x [0,0] x [0,1] + [0,1] x [0,1] x [0,0]
- [0,1] x [0,1] x [1,1] + [0,1] x [1,1] x [0,1] - [1,1] x [0,1] x [0,1])]

Similarly for simplicial sets::

sage: S = simplicial_sets.Sphere(2)
sage: S.homology(generators=True) # needs sage.modules
sage: S.homology(generators=True)
{0: [], 1: 0, 2: [(Z, sigma_2)]}
"""
from sage.homology.homology_group import HomologyGroup
Expand Down Expand Up @@ -625,14 +622,13 @@ def cohomology(self, dim=None, base_ring=ZZ, subcomplex=None,
EXAMPLES::

sage: circle = SimplicialComplex([[0,1], [1,2], [0, 2]])
sage: circle.cohomology(0) # needs sage.modules
sage: circle.cohomology(0)
0
sage: circle.cohomology(1) # needs sage.modules
sage: circle.cohomology(1)
Z

Projective plane::

sage: # needs sage.modules
sage: P2 = SimplicialComplex([[0,1,2], [0,2,3], [0,1,5], [0,4,5], [0,3,4],
....: [1,2,4], [1,3,4], [1,3,5], [2,3,5], [2,4,5]])
sage: P2.cohomology(2)
Expand All @@ -642,20 +638,20 @@ def cohomology(self, dim=None, base_ring=ZZ, subcomplex=None,
sage: P2.cohomology(2, base_ring=GF(3))
Vector space of dimension 0 over Finite Field of size 3

sage: cubical_complexes.KleinBottle().cohomology(2) # needs sage.modules
sage: cubical_complexes.KleinBottle().cohomology(2)
C2

Relative cohomology::

sage: T = SimplicialComplex([[0,1]])
sage: U = SimplicialComplex([[0], [1]])
sage: T.cohomology(1, subcomplex=U) # needs sage.modules
sage: T.cohomology(1, subcomplex=U)
Z

A `\Delta`-complex example::

sage: s5 = delta_complexes.Sphere(5)
sage: s5.cohomology(base_ring=GF(7))[5] # needs sage.modules
sage: s5.cohomology(base_ring=GF(7))[5]
Vector space of dimension 1 over Finite Field of size 7
"""
return self.homology(dim=dim, cohomology=True, base_ring=base_ring,
Expand Down Expand Up @@ -689,23 +685,23 @@ def betti(self, dim=None, subcomplex=None):
two-point space with itself::

sage: S = SimplicialComplex([[0], [1]])
sage: (S*S*S).betti() # needs sage.modules
sage: (S*S*S).betti()
{0: 1, 1: 0, 2: 1}
sage: (S*S*S).betti([1,2]) # needs sage.modules
sage: (S*S*S).betti([1,2])
{1: 0, 2: 1}
sage: (S*S*S).betti(2) # needs sage.modules
sage: (S*S*S).betti(2)
1

Or build the two-sphere as a `\Delta`-complex::

sage: S2 = delta_complexes.Sphere(2)
sage: S2.betti([1,2]) # needs sage.modules
sage: S2.betti([1,2])
{1: 0, 2: 1}

Or as a cubical complex::

sage: S2c = cubical_complexes.Sphere(2)
sage: S2c.betti(2) # needs sage.modules
sage: S2c.betti(2)
1
"""
dic = {}
Expand Down Expand Up @@ -733,9 +729,9 @@ def is_acyclic(self, base_ring=ZZ) -> bool:
EXAMPLES::

sage: RP2 = simplicial_complexes.RealProjectivePlane()
sage: RP2.is_acyclic() # needs sage.modules
sage: RP2.is_acyclic()
False
sage: RP2.is_acyclic(QQ) # needs sage.modules
sage: RP2.is_acyclic(QQ)
True

This first computes the Euler characteristic: if it is not 1,
Expand Down Expand Up @@ -779,12 +775,12 @@ def n_chains(self, n, base_ring=ZZ, cochains=False):
EXAMPLES::

sage: S2 = simplicial_complexes.Sphere(2)
sage: S2.n_chains(1, QQ) # needs sage.modules
sage: S2.n_chains(1, QQ)
Free module generated by {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)}
over Rational Field
sage: list(S2.n_chains(1, QQ, cochains=False).basis()) # needs sage.modules
sage: list(S2.n_chains(1, QQ, cochains=False).basis())
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
sage: list(S2.n_chains(1, QQ, cochains=True).basis()) # needs sage.modules
sage: list(S2.n_chains(1, QQ, cochains=True).basis())
[\chi_(0, 1), \chi_(0, 2), \chi_(0, 3), \chi_(1, 2), \chi_(1, 3), \chi_(2, 3)]
"""
from sage.homology.chains import Chains, Cochains
Expand Down Expand Up @@ -847,7 +843,6 @@ def homology_with_basis(self, base_ring=QQ, cohomology=False):

EXAMPLES::

sage: # needs sage.modules
sage: K = simplicial_complexes.KleinBottle()
sage: H = K.homology_with_basis(QQ); H
Homology module of Minimal triangulation of the Klein bottle
Expand All @@ -863,12 +858,12 @@ def homology_with_basis(self, base_ring=QQ, cohomology=False):
The homology is constructed as a graded object, so for
example, you can ask for the basis in a single degree::

sage: H.basis(1) # needs sage.modules
sage: H.basis(1)
Finite family {(1, 0): h_{1,0}, (1, 1): h_{1,1}}

sage: S3 = delta_complexes.Sphere(3)
sage: H = S3.homology_with_basis(QQ, cohomology=True) # needs sage.modules
sage: list(H.basis(3)) # needs sage.modules
sage: H = S3.homology_with_basis(QQ, cohomology=True)
sage: list(H.basis(3))
[h^{3,0}]
"""
from sage.homology.homology_vector_space_with_basis import \
Expand Down Expand Up @@ -914,7 +909,6 @@ def cohomology_ring(self, base_ring=QQ):

EXAMPLES::

sage: # needs sage.modules
sage: K = simplicial_complexes.KleinBottle()
sage: H = K.cohomology_ring(QQ); H
Cohomology ring of Minimal triangulation of the Klein bottle
Expand All @@ -928,23 +922,22 @@ def cohomology_ring(self, base_ring=QQ):
[h^{0,0}, h^{1,0}, h^{1,1}, h^{2,0}]

sage: X = delta_complexes.SurfaceOfGenus(2)
sage: H = X.cohomology_ring(QQ); H # needs sage.modules
sage: H = X.cohomology_ring(QQ); H
Cohomology ring of Delta complex with 3 vertices and 29 simplices
over Rational Field
sage: sorted(H.basis(1), key=str) # needs sage.modules
sage: sorted(H.basis(1), key=str)
[h^{1,0}, h^{1,1}, h^{1,2}, h^{1,3}]

sage: H = simplicial_complexes.Torus().cohomology_ring(QQ); H # needs sage.modules
sage: H = simplicial_complexes.Torus().cohomology_ring(QQ); H
Cohomology ring of Minimal triangulation of the torus
over Rational Field
sage: x = H.basis()[1,0]; x # needs sage.modules
sage: x = H.basis()[1,0]; x
h^{1,0}
sage: y = H.basis()[1,1]; y # needs sage.modules
sage: y = H.basis()[1,1]; y
h^{1,1}

You can compute cup products of cohomology classes::

sage: # needs sage.modules
sage: x.cup_product(y)
-h^{2,0}
sage: x * y # alternate notation
Expand All @@ -956,13 +949,12 @@ def cohomology_ring(self, base_ring=QQ):

Cohomology operations::

sage: # needs sage.groups
sage: RP2 = simplicial_complexes.RealProjectivePlane()
sage: K = RP2.suspension()
sage: K.set_immutable()
sage: y = K.cohomology_ring(GF(2)).basis()[2,0]; y # needs sage.modules
sage: y = K.cohomology_ring(GF(2)).basis()[2,0]; y
h^{2,0}
sage: y.Sq(1) # needs sage.modules
sage: y.Sq(1)
h^{3,0}

To compute the cohomology ring, the complex must be
Expand All @@ -976,12 +968,12 @@ def cohomology_ring(self, base_ring=QQ):
sage: T = S1.product(S1)
sage: T.is_immutable()
False
sage: T.cohomology_ring() # needs sage.modules
sage: T.cohomology_ring()
Traceback (most recent call last):
...
ValueError: this simplicial complex must be immutable; call set_immutable()
sage: T.set_immutable()
sage: T.cohomology_ring() # needs sage.modules
sage: T.cohomology_ring()
Cohomology ring of Simplicial complex with 9 vertices and
18 facets over Rational Field
"""
Expand Down
34 changes: 15 additions & 19 deletions src/sage/topology/cubical_complex.py
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Original file line number Diff line number Diff line change
@@ -1,4 +1,3 @@
# sage.doctest: needs sage.graphs
r"""
Finite cubical complexes

Expand Down Expand Up @@ -46,11 +45,11 @@
segment in the plane from `(0,2)` to `(0,3)`. We could form a
topologically equivalent space by inserting some degenerate simplices::

sage: S1.homology() # needs sage.modules
sage: S1.homology()
{0: 0, 1: Z}
sage: X = CubicalComplex([([0,0], [2,3], [2]), ([0,1], [3,3], [2]),
....: ([0,1], [2,2], [2]), ([1,1], [2,3], [2])])
sage: X.homology() # needs sage.modules
sage: X.homology()
{0: 0, 1: Z}

Topologically, the cubical complex ``X`` consists of four edges of a
Expand Down Expand Up @@ -797,26 +796,26 @@ class :class:`Cube`, or lists or tuples suitable for conversion to
sage: S1 = CubicalComplex([([0,0], [2,3]), ([0,1], [3,3]),
....: ([0,1], [2,2]), ([1,1], [2,3])]); S1
Cubical complex with 4 vertices and 8 cubes
sage: S1.homology() # needs sage.modules
sage: S1.homology()
{0: 0, 1: Z}

A set of five points and its product with ``S1``::

sage: pts = CubicalComplex([([0],), ([3],), ([6],), ([-12],), ([5],)])
sage: pts
Cubical complex with 5 vertices and 5 cubes
sage: pts.homology() # needs sage.modules
sage: pts.homology()
{0: Z x Z x Z x Z}
sage: X = S1.product(pts); X
Cubical complex with 20 vertices and 40 cubes
sage: X.homology() # needs sage.modules
sage: X.homology()
{0: Z x Z x Z x Z, 1: Z^5}

Converting a simplicial complex to a cubical complex::

sage: S2 = simplicial_complexes.Sphere(2)
sage: C2 = CubicalComplex(S2)
sage: all(C2.homology(n) == S2.homology(n) for n in range(3)) # needs sage.modules
sage: all(C2.homology(n) == S2.homology(n) for n in range(3))
True

You can get the set of maximal cells or a dictionary of all cells::
Expand Down Expand Up @@ -853,14 +852,14 @@ class :class:`Cube`, or lists or tuples suitable for conversion to

sage: T = S1.product(S1); T
Cubical complex with 16 vertices and 64 cubes
sage: T.chain_complex() # needs sage.modules
sage: T.chain_complex()
Chain complex with at most 3 nonzero terms over Integer Ring
sage: T.homology(base_ring=QQ) # needs sage.modules
sage: T.homology(base_ring=QQ)
{0: Vector space of dimension 0 over Rational Field,
1: Vector space of dimension 2 over Rational Field,
2: Vector space of dimension 1 over Rational Field}
sage: RP2 = cubical_complexes.RealProjectivePlane()
sage: RP2.cohomology(dim=[1, 2], base_ring=GF(2)) # needs sage.modules
sage: RP2.cohomology(dim=[1, 2], base_ring=GF(2))
{1: Vector space of dimension 1 over Finite Field of size 2,
2: Vector space of dimension 1 over Finite Field of size 2}

Expand Down Expand Up @@ -1191,7 +1190,6 @@ def chain_complex(self, subcomplex=None, augmented=False,

EXAMPLES::

sage: # needs sage.modules
sage: S2 = cubical_complexes.Sphere(2)
sage: S2.chain_complex()
Chain complex with at most 3 nonzero terms over Integer Ring
Expand All @@ -1210,7 +1208,6 @@ def chain_complex(self, subcomplex=None, augmented=False,

Check that :issue:`32203` has been fixed::

sage: # needs sage.modules
sage: Square = CubicalComplex([([0,1],[0,1])])
sage: EdgesLTR = CubicalComplex([([0,0],[0,1]),([0,1],[1,1]),([1,1],[0,1])])
sage: EdgesLBR = CubicalComplex([([0,0],[0,1]),([0,1],[0,0]),([1,1],[0,1])])
Expand Down Expand Up @@ -1507,7 +1504,7 @@ def disjoint_union(self, other):

sage: S1 = cubical_complexes.Sphere(1)
sage: S2 = cubical_complexes.Sphere(2)
sage: S1.disjoint_union(S2).homology() # needs sage.modules
sage: S1.disjoint_union(S2).homology()
{0: Z, 1: Z, 2: Z}
"""
embedded_left = len(tuple(self.maximal_cells()[0]))
Expand Down Expand Up @@ -1545,7 +1542,7 @@ def wedge(self, other):

sage: S1 = cubical_complexes.Sphere(1)
sage: S2 = cubical_complexes.Sphere(2)
sage: S1.wedge(S2).homology() # needs sage.modules
sage: S1.wedge(S2).homology()
{0: 0, 1: Z, 2: Z}
"""
embedded_left = len(tuple(self.maximal_cells()[0]))
Expand Down Expand Up @@ -1582,12 +1579,12 @@ def connected_sum(self, other):

sage: T = cubical_complexes.Torus()
sage: S2 = cubical_complexes.Sphere(2)
sage: T.connected_sum(S2).cohomology() == T.cohomology() # needs sage.modules
sage: T.connected_sum(S2).cohomology() == T.cohomology()
True
sage: RP2 = cubical_complexes.RealProjectivePlane()
sage: T.connected_sum(RP2).homology(1) # needs sage.modules
sage: T.connected_sum(RP2).homology(1)
Z x Z x C2
sage: RP2.connected_sum(RP2).connected_sum(RP2).homology(1) # needs sage.modules
sage: RP2.connected_sum(RP2).connected_sum(RP2).homology(1)
Z x Z x C2
"""
# connected_sum: first check whether the complexes are pure
Expand Down Expand Up @@ -1703,7 +1700,6 @@ def algebraic_topological_model(self, base_ring=None):

EXAMPLES::

sage: # needs sage.modules
sage: RP2 = cubical_complexes.RealProjectivePlane()
sage: phi, M = RP2.algebraic_topological_model(GF(2))
sage: M.homology()
Expand Down Expand Up @@ -1764,7 +1760,7 @@ def _simplicial_(self):

sage: Ts = T._simplicial_(); Ts
Simplicial complex with 16 vertices and 32 facets
sage: T.homology() == Ts.homology() # needs sage.modules
sage: T.homology() == Ts.homology()
True

Each `n`-dimensional cube produces `n!` `n`-simplices::
Expand Down
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