def Sphere(self,n):
"""
A minimal triangulation of the n-dimensional sphere.
INPUT:
- ``n`` - positive integer
EXAMPLES::
sage: simplicial_complexes.Sphere(2)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
sage: simplicial_complexes.Sphere(5).homology()
{0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: Z}
sage: [simplicial_complexes.Sphere(n).euler_characteristic() for n in range(6)]
[2, 0, 2, 0, 2, 0]
sage: [simplicial_complexes.Sphere(n).f_vector() for n in range(6)]
[[1, 2],
[1, 3, 3],
[1, 4, 6, 4],
[1, 5, 10, 10, 5],
[1, 6, 15, 20, 15, 6],
[1, 7, 21, 35, 35, 21, 7]]
"""
S = Simplex(n+1)
facets = S.faces()
S = SimplicialComplex(n+1, facets)
return S
def Sphere(self, n):
"""
A minimal triangulation of the n-dimensional sphere.
INPUT:
- ``n`` - positive integer
EXAMPLES::
sage: simplicial_complexes.Sphere(2)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
sage: simplicial_complexes.Sphere(5).homology()
{0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: Z}
sage: [simplicial_complexes.Sphere(n).euler_characteristic() for n in range(6)]
[2, 0, 2, 0, 2, 0]
sage: [simplicial_complexes.Sphere(n).f_vector() for n in range(6)]
[[1, 2],
[1, 3, 3],
[1, 4, 6, 4],
[1, 5, 10, 10, 5],
[1, 6, 15, 20, 15, 6],
[1, 7, 21, 35, 35, 21, 7]]
"""
S = Simplex(n + 1)
facets = S.faces()
S = SimplicialComplex(n + 1, facets)
return S
def __call__(self, x, orientation=False):
"""
Input is a simplex of the domain. Output is the image simplex.
If the optional argument ``orientation`` is ``True``, then this
returns a pair ``(image simplex, oriented)`` where ``oriented``
is 1 or `-1` depending on whether the map preserves or reverses
the orientation of the image simplex.
EXAMPLES::
sage: S = simplicial_complexes.Sphere(2)
sage: T = simplicial_complexes.Sphere(3)
sage: S
Minimal triangulation of the 2-sphere
sage: T
Minimal triangulation of the 3-sphere
sage: f = {0:0,1:1,2:2,3:3}
sage: H = Hom(S,T)
sage: x = H(f)
sage: from sage.homology.simplicial_complex import Simplex
sage: x(Simplex([0,2,3]))
(0, 2, 3)
An orientation-reversing example::
sage: X = SimplicialComplex([[0,1]], is_mutable=False)
sage: g = Hom(X,X)({0:1, 1:0})
sage: g(Simplex([0,1]))
(0, 1)
sage: g(Simplex([0,1]), orientation=True)
((0, 1), -1)
"""
dim = self.domain().dimension()
if not isinstance(x, Simplex) or x.dimension(
) > dim or x not in self.domain().faces()[x.dimension()]:
raise ValueError("x must be a simplex of the source of f")
tup = x.tuple()
fx = []
for j in tup:
fx.append(self._vertex_dictionary[j])
if orientation:
if len(set(fx)) == len(tup):
oriented = Permutation(convert_perm(fx)).signature()
else:
oriented = 1
return (Simplex(set(fx)), oriented)
else:
return Simplex(set(fx))
def fiber_product(self, other, rename_vertices=True):
"""
Fiber product of ``self`` and ``other``. Both morphisms should have
the same codomain. The method returns a morphism of simplicial
complexes, which is the morphism from the space of the fiber product
to the codomain.
EXAMPLES::
sage: S = SimplicialComplex([[0,1],[1,2]], is_mutable=False)
sage: T = SimplicialComplex([[0,2],[1]], is_mutable=False)
sage: U = SimplicialComplex([[0,1],[2]], is_mutable=False)
sage: H = Hom(S,U)
sage: G = Hom(T,U)
sage: f = {0:0,1:1,2:0}
sage: g = {0:0,1:1,2:1}
sage: x = H(f)
sage: y = G(g)
sage: z = x.fiber_product(y)
sage: z
Simplicial complex morphism:
From: Simplicial complex with 4 vertices and facets {...}
To: Simplicial complex with vertex set (0, 1, 2) and facets {(2,), (0, 1)}
Defn: L0R0 |--> 0
L1R1 |--> 1
L1R2 |--> 1
L2R0 |--> 0
"""
if self.codomain() != other.codomain():
raise ValueError("self and other must have the same codomain.")
X = self.domain().product(other.domain(),
rename_vertices=rename_vertices)
v = []
f = dict()
eff1 = self.domain().vertices()
eff2 = other.domain().vertices()
for i in eff1:
for j in eff2:
if self(Simplex([i])) == other(Simplex([j])):
if rename_vertices:
v.append("L" + str(i) + "R" + str(j))
f["L" + str(i) + "R" +
str(j)] = self._vertex_dictionary[i]
else:
v.append((i, j))
f[(i, j)] = self._vertex_dictionary[i]
return SimplicialComplexMorphism(f, X.generated_subcomplex(v),
self.codomain())
def __init__(self, f, X, Y):
"""
Input is a dictionary ``f``, the domain ``X``, and the codomain ``Y``.
One can define the dictionary on the vertices of `X`.
EXAMPLES::
sage: S = SimplicialComplex([[0,1],[2],[3,4],[5]], is_mutable=False)
sage: H = Hom(S,S)
sage: f = {0:0,1:1,2:2,3:3,4:4,5:5}
sage: g = {0:0,1:1,2:0,3:3,4:4,5:0}
sage: x = H(f)
sage: y = H(g)
sage: x == y
False
sage: x.image()
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(2,), (5,), (0, 1), (3, 4)}
sage: y.image()
Simplicial complex with vertex set (0, 1, 3, 4) and facets {(0, 1), (3, 4)}
sage: x.image() == y.image()
False
"""
if not isinstance(X, SimplicialComplex) or not isinstance(
Y, SimplicialComplex):
raise ValueError("X and Y must be SimplicialComplexes")
if not set(f.keys()) == set(X.vertices()):
raise ValueError(
"f must be a dictionary from the vertex set of X to single values in the vertex set of Y"
)
dim = X.dimension()
Y_faces = Y.faces()
for k in range(dim + 1):
for i in X.faces()[k]:
tup = i.tuple()
fi = []
for j in tup:
fi.append(f[j])
v = Simplex(set(fi))
if v not in Y_faces[v.dimension()]:
raise ValueError(
"f must be a dictionary from the vertices of X to the vertices of Y"
)
self._vertex_dictionary = f
Morphism.__init__(self, Hom(X, Y, SimplicialComplexes()))
def __init__(self,f,X,Y):
"""
Input is a dictionary ``f``, the domain ``X``, and the codomain ``Y``.
One can define the dictionary on the vertices of `X`.
EXAMPLES::
sage: S = SimplicialComplex([[0,1],[2],[3,4],[5]], is_mutable=False)
sage: H = Hom(S,S)
sage: f = {0:0,1:1,2:2,3:3,4:4,5:5}
sage: g = {0:0,1:1,2:0,3:3,4:4,5:0}
sage: x = H(f)
sage: y = H(g)
sage: x == y
False
sage: x.image()
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(3, 4), (5,), (2,), (0, 1)}
sage: y.image()
Simplicial complex with vertex set (0, 1, 3, 4) and facets {(3, 4), (0, 1)}
sage: x.image() == y.image()
False
"""
if not isinstance(X,SimplicialComplex) or not isinstance(Y,SimplicialComplex):
raise ValueError("X and Y must be SimplicialComplexes")
if not set(f.keys()) == set(X._vertex_set):
raise ValueError("f must be a dictionary from the vertex set of X to single values in the vertex set of Y")
dim = X.dimension()
Y_faces = Y.faces()
for k in range(dim+1):
for i in X.faces()[k]:
tup = i.tuple()
fi = []
for j in tup:
fi.append(f[j])
v = Simplex(set(fi))
if not v in Y_faces[v.dimension()]:
raise ValueError("f must be a dictionary from the vertices of X to the vertices of Y")
self._vertex_dictionary = f
Morphism.__init__(self, Hom(X,Y,SimplicialComplexes()))
def Simplex(self, n):
"""
An `n`-dimensional simplex, as a simplicial complex.
INPUT:
- ``n`` - a non-negative integer
OUTPUT: the simplicial complex consisting of the `n`-simplex
on vertices `(0, 1, ..., n)` and all of its faces.
EXAMPLES::
sage: simplicial_complexes.Simplex(3)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 2, 3)}
sage: simplicial_complexes.Simplex(5).euler_characteristic()
1
"""
S = Simplex(n)
return SimplicialComplex(n, list(S))
def is_contiguous_to(self, other):
r"""
Return ``True`` if ``self`` is contiguous to ``other``.
Two morphisms `f_0, f_1: K \to L` are *contiguous* if for any
simplex `\sigma \in K`, the union `f_0(\sigma) \cup
f_1(\sigma)` is a simplex in `L`. This is not a transitive
relation, but it induces an equivalence relation on simplicial
maps: `f` is equivalent to `g` if there is a finite sequence
`f_0 = f`, `f_1`, ..., `f_n = g` such that `f_i` and `f_{i+1}`
are contiguous for each `i`.
This is related to maps being homotopic: if they are
contiguous, then they induce homotopic maps on the geometric
realizations. Given two homotopic maps on the geometric
realizations, then after barycentrically subdividing `n` times
for some `n`, the maps have simplicial approximations which
are in the same contiguity class. (This last fact is only true
if the domain is a *finite* simplicial complex, by the way.)
See Section 3.5 of Spanier [Spa1966]_ for details.
ALGORITHM:
It is enough to check when `\sigma` ranges over the facets.
INPUT:
- ``other`` -- a simplicial complex morphism with the same
domain and codomain as ``self``
EXAMPLES::
sage: K = simplicial_complexes.Simplex(1)
sage: L = simplicial_complexes.Sphere(1)
sage: H = Hom(K, L)
sage: f = H({0: 0, 1: 1})
sage: g = H({0: 0, 1: 0})
sage: f.is_contiguous_to(f)
True
sage: f.is_contiguous_to(g)
True
sage: h = H({0: 1, 1: 2})
sage: f.is_contiguous_to(h)
False
TESTS::
sage: one = Hom(K,K).identity()
sage: one.is_contiguous_to(f)
False
sage: one.is_contiguous_to(3) # nonsensical input
False
"""
if not isinstance(other, SimplicialComplexMorphism):
return False
if self.codomain() != other.codomain() or self.domain() != other.domain():
return False
domain = self.domain()
codomain = self.codomain()
return all(Simplex(self(sigma).set().union(other(sigma))) in codomain
for sigma in domain.facets())