def fundamental_group(self, simplify=True):
r"""
Return the fundamental group of this pointed simplicial set.
INPUT:
- ``simplify`` (bool, optional ``True``) -- if
``False``, then return a presentation of the group
in terms of generators and relations. If ``True``,
the default, simplify as much as GAP is able to.
Algorithm: we compute the edge-path group -- see
Section 19 of [Kan1958]_ and
:wikipedia:`Fundamental_group`. Choose a spanning tree
for the connected component of the 1-skeleton
containing the base point, and then the group's
generators are given by the non-degenerate
edges. There are two types of relations: `e=1` if `e`
is in the spanning tree, and for every 2-simplex, if
its faces are `e_0`, `e_1`, and `e_2`, then we impose
the relation `e_0 e_1^{-1} e_2 = 1`, where we first
set `e_i=1` if `e_i` is degenerate.
EXAMPLES::
sage: S1 = simplicial_sets.Sphere(1)
sage: eight = S1.wedge(S1)
sage: eight.fundamental_group() # free group on 2 generators
Finitely presented group < e0, e1 | >
The fundamental group of a disjoint union of course depends on
the choice of base point::
sage: T = simplicial_sets.Torus()
sage: K = simplicial_sets.KleinBottle()
sage: X = T.disjoint_union(K)
sage: X_0 = X.set_base_point(X.n_cells(0)[0])
sage: X_0.fundamental_group().is_abelian()
True
sage: X_1 = X.set_base_point(X.n_cells(0)[1])
sage: X_1.fundamental_group().is_abelian()
False
sage: RP3 = simplicial_sets.RealProjectiveSpace(3)
sage: RP3.fundamental_group()
Finitely presented group < e | e^2 >
Compute the fundamental group of some classifying spaces::
sage: C5 = groups.misc.MultiplicativeAbelian([5])
sage: BC5 = C5.nerve()
sage: BC5.fundamental_group()
Finitely presented group < e0 | e0^5 >
sage: Sigma3 = groups.permutation.Symmetric(3)
sage: BSigma3 = Sigma3.nerve()
sage: pi = BSigma3.fundamental_group(); pi
Finitely presented group < e0, e1 | e0^2, e1^3, (e0*e1^-1)^2 >
sage: pi.order()
6
sage: pi.is_abelian()
False
The sphere has a trivial fundamental group::
sage: S2 = simplicial_sets.Sphere(2)
sage: S2.fundamental_group()
Finitely presented group < | >
"""
# Import this here to prevent importing libgap upon startup.
from sage.groups.free_group import FreeGroup
skel = self.n_skeleton(2)
graph = skel.graph()
if not skel.is_connected():
graph = graph.subgraph(skel.base_point())
edges = [e[2] for e in graph.edges()]
spanning_tree = [e[2] for e in graph.min_spanning_tree()]
gens = [e for e in edges if e not in spanning_tree]
if not gens:
return FreeGroup([]).quotient([])
gens_dict = dict(zip(gens, range(len(gens))))
FG = FreeGroup(len(gens), 'e')
rels = []
for f in skel.n_cells(2):
z = dict()
for i, sigma in enumerate(skel.faces(f)):
if sigma in spanning_tree:
z[i] = FG.one()
elif sigma.is_degenerate():
z[i] = FG.one()
elif sigma in edges:
z[i] = FG.gen(gens_dict[sigma])
else:
# sigma is not in the correct connected component.
z[i] = FG.one()
rels.append(z[0] * z[1].inverse() * z[2])
if simplify:
return FG.quotient(rels).simplified()
else:
return FG.quotient(rels)
def fundamental_group(self, simplify=True):
r"""
Return the fundamental group of this pointed simplicial set.
INPUT:
- ``simplify`` (bool, optional ``True``) -- if
``False``, then return a presentation of the group
in terms of generators and relations. If ``True``,
the default, simplify as much as GAP is able to.
Algorithm: we compute the edge-path group -- see
Section 19 of [Kan1958]_ and
:wikipedia:`Fundamental_group`. Choose a spanning tree
for the connected component of the 1-skeleton
containing the base point, and then the group's
generators are given by the non-degenerate
edges. There are two types of relations: `e=1` if `e`
is in the spanning tree, and for every 2-simplex, if
its faces are `e_0`, `e_1`, and `e_2`, then we impose
the relation `e_0 e_1^{-1} e_2 = 1`, where we first
set `e_i=1` if `e_i` is degenerate.
EXAMPLES::
sage: S1 = simplicial_sets.Sphere(1)
sage: eight = S1.wedge(S1)
sage: eight.fundamental_group() # free group on 2 generators
Finitely presented group < e0, e1 | >
The fundamental group of a disjoint union of course depends on
the choice of base point::
sage: T = simplicial_sets.Torus()
sage: K = simplicial_sets.KleinBottle()
sage: X = T.disjoint_union(K)
sage: X_0 = X.set_base_point(X.n_cells(0)[0])
sage: X_0.fundamental_group().is_abelian()
True
sage: X_1 = X.set_base_point(X.n_cells(0)[1])
sage: X_1.fundamental_group().is_abelian()
False
sage: RP3 = simplicial_sets.RealProjectiveSpace(3)
sage: RP3.fundamental_group()
Finitely presented group < e | e^2 >
Compute the fundamental group of some classifying spaces::
sage: C5 = groups.misc.MultiplicativeAbelian([5])
sage: BC5 = C5.nerve()
sage: BC5.fundamental_group()
Finitely presented group < e0 | e0^5 >
sage: Sigma3 = groups.permutation.Symmetric(3)
sage: BSigma3 = Sigma3.nerve()
sage: pi = BSigma3.fundamental_group(); pi
Finitely presented group < e0, e1 | e0^2, e1^3, (e0*e1^-1)^2 >
sage: pi.order()
6
sage: pi.is_abelian()
False
"""
# Import this here to prevent importing libgap upon startup.
from sage.groups.free_group import FreeGroup
skel = self.n_skeleton(2)
graph = skel.graph()
if not skel.is_connected():
graph = graph.subgraph(skel.base_point())
edges = [e[2] for e in graph.edges()]
spanning_tree = [e[2] for e in graph.min_spanning_tree()]
gens = [e for e in edges if e not in spanning_tree]
if not gens:
return gap.TrivialGroup()
gens_dict = dict(zip(gens, range(len(gens))))
FG = FreeGroup(len(gens), 'e')
rels = []
for f in skel.n_cells(2):
z = dict()
for i, sigma in enumerate(skel.faces(f)):
if sigma in spanning_tree:
z[i] = FG.one()
elif sigma.is_degenerate():
z[i] = FG.one()
elif sigma in edges:
z[i] = FG.gen(gens_dict[sigma])
else:
# sigma is not in the correct connected component.
z[i] = FG.one()
rels.append(z[0]*z[1].inverse()*z[2])
if simplify:
return FG.quotient(rels).simplified()
else:
return FG.quotient(rels)
