def fundamental_group(f, simplified=True, projective=False):
r"""
Return a presentation of the fundamental group of the complement of
the algebraic set defined by the polynomial ``f``.
INPUT:
- ``f`` -- a polynomial in two variables, with coefficients in either
the rationals or a number field with a fixed embedding in `\QQbar`
- ``simplified`` -- boolean (default: ``True``); if set to ``True`` the
presentation will be simplified (see below)
- ``projective`` -- boolean (default: ``False``); if set to ``True``,
the fundamental group of the complement of the projective completion
of the curve will be computed, otherwise, the fundamental group of
the complement in the affine plane will be computed
If ``simplified`` is ``False``, the returned presentation has as
many generators as degree of the polynomial times the points in the
base used to create the segments that surround the discriminant. In
this case, the generators are granted to be meridians of the curve.
OUTPUT:
A presentation of the fundamental group of the complement of the
curve defined by ``f``.
EXAMPLES::
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
sage: R.<x,y> = QQ[]
sage: f = x^2 + y^3
sage: fundamental_group(f) # optional - sirocco
Finitely presented group < ... >
sage: fundamental_group(f, simplified=False) # optional - sirocco
Finitely presented group < ... >
::
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
sage: R.<x,y> = QQ[]
sage: f = y^3 + x^3
sage: fundamental_group(f) # optional - sirocco
Finitely presented group < ... >
It is also possible to have coefficients in a number field with a
fixed embedding in `\QQbar`::
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
sage: zeta = QQbar['x']('x^2+x+1').roots(multiplicities=False)[0]
sage: zeta
-0.50000000000000000? - 0.866025403784439?*I
sage: F = NumberField(zeta.minpoly(), 'zeta', embedding=zeta)
sage: F.inject_variables()
Defining zeta
sage: R.<x,y> = F[]
sage: f = y^3 + x^3 +zeta *x + 1
sage: fundamental_group(f) # optional - sirocco
Finitely presented group < x0 | >
"""
(x, y) = f.variables()
F = f.base_ring()
g = f.factor().radical().prod()
d = g.degree(y)
while not g.coefficient(y**d) in F or (projective and g.total_degree() > d):
g = g.subs({x: x + y})
d = g.degree(y)
disc = discrim(g)
segs = segments(disc)
vertices = list(set(flatten(segs)))
Faux = FreeGroup(d)
F = FreeGroup(d * len(vertices))
rels = []
if projective:
rels.append(prod(F.gen(i) for i in range(d)))
braidscomputed = braid_in_segment([(g, seg[0], seg[1]) for seg in segs])
for braidcomputed in braidscomputed:
seg = (braidcomputed[0][0][1], braidcomputed[0][0][2])
b = braidcomputed[1]
i = vertices.index(seg[0])
j = vertices.index(seg[1])
for k in range(d):
el1 = Faux([k + 1]) * b.inverse()
el2 = k + 1
w1 = F([sign(a)*d*i + a for a in el1.Tietze()])
w2 = F([d*j + el2])
rels.append(w1/w2)
G = F / rels
if simplified:
return G.simplified()
else:
return G
def FinitelyGeneratedAbelianPresentation(int_list):
r"""
Return canonical presentation of finitely generated abelian group.
INPUT:
- ``int_list`` -- List of integers defining the group to be returned, the defining list
is reduced to the invariants of the input list before generating the corresponding
group.
OUTPUT:
Finitely generated abelian group, `\ZZ_{n_1} \times \ZZ_{n_2} \times \cdots \times \ZZ_{n_k}`
as a finite presentation, where `n_i` forms the invariants of the input list.
EXAMPLES::
sage: groups.presentation.FGAbelian([2,2])
Finitely presented group < a, b | a^2, b^2, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([2,3])
Finitely presented group < a | a^6 >
sage: groups.presentation.FGAbelian([2,4])
Finitely presented group < a, b | a^2, b^4, a^-1*b^-1*a*b >
You can create free abelian groups::
sage: groups.presentation.FGAbelian([0])
Finitely presented group < a | >
sage: groups.presentation.FGAbelian([0,0])
Finitely presented group < a, b | a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([0,0,0])
Finitely presented group < a, b, c | a^-1*b^-1*a*b, a^-1*c^-1*a*c, b^-1*c^-1*b*c >
And various infinite abelian groups::
sage: groups.presentation.FGAbelian([0,2])
Finitely presented group < a, b | a^2, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([0,2,2])
Finitely presented group < a, b, c | a^2, b^2, a^-1*b^-1*a*b, a^-1*c^-1*a*c, b^-1*c^-1*b*c >
Outputs are reduced to minimal generators and relations::
sage: groups.presentation.FGAbelian([3,5,2,7,3])
Finitely presented group < a, b | a^3, b^210, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([3,210])
Finitely presented group < a, b | a^3, b^210, a^-1*b^-1*a*b >
The trivial group is an acceptable output::
sage: groups.presentation.FGAbelian([])
Finitely presented group < | >
sage: groups.presentation.FGAbelian([1])
Finitely presented group < | >
sage: groups.presentation.FGAbelian([1,1,1,1,1,1,1,1,1,1])
Finitely presented group < | >
Input list must consist of positive integers::
sage: groups.presentation.FGAbelian([2,6,3,9,-4])
Traceback (most recent call last):
...
ValueError: input list must contain nonnegative entries
sage: groups.presentation.FGAbelian([2,'a',4])
Traceback (most recent call last):
...
TypeError: unable to convert 'a' to an integer
TESTS::
sage: ag = groups.presentation.FGAbelian([2,2])
sage: ag.as_permutation_group().is_isomorphic(groups.permutation.KleinFour())
True
sage: G = groups.presentation.FGAbelian([2,4,8])
sage: C2 = CyclicPermutationGroup(2)
sage: C4 = CyclicPermutationGroup(4)
sage: C8 = CyclicPermutationGroup(8)
sage: gg = (C2.direct_product(C4)[0]).direct_product(C8)[0]
sage: gg.is_isomorphic(G.as_permutation_group())
True
sage: all(groups.presentation.FGAbelian([i]).as_permutation_group().is_isomorphic(groups.presentation.Cyclic(i).as_permutation_group()) for i in [2..35])
True
"""
from sage.groups.free_group import _lexi_gen
check_ls = [Integer(x) for x in int_list if Integer(x) >= 0]
if len(check_ls) != len(int_list):
raise ValueError('input list must contain nonnegative entries')
col_sp = diagonal_matrix(int_list).column_space()
invariants = FGP_Module(ZZ**(len(int_list)), col_sp).invariants()
name_gen = _lexi_gen()
F = FreeGroup([next(name_gen) for i in invariants])
ret_rls = [
F([i + 1])**invariants[i] for i in range(len(invariants))
if invariants[i] != 0
]
# Build commutator relations
gen_pairs = [[F.gen(i), F.gen(j)] for i in range(F.ngens() - 1)
for j in range(i + 1, F.ngens())]
ret_rls = ret_rls + [
x[0]**(-1) * x[1]**(-1) * x[0] * x[1] for x in gen_pairs
]
return FinitelyPresentedGroup(F, tuple(ret_rls))
def FinitelyGeneratedHeisenbergPresentation(n=1, p=0):
r"""
Return a finite presentation of the Heisenberg group.
The Heisenberg group is the group of `(n+2) \times (n+2)` matrices
over a ring `R` with diagonal elements equal to 1, first row and
last column possibly nonzero, and all the other entries equal to zero.
INPUT:
- ``n`` -- the degree of the Heisenberg group
- ``p`` -- (optional) a prime number, where we construct the
Heisenberg group over the finite field `\ZZ/p\ZZ`
OUTPUT:
Finitely generated Heisenberg group over the finite field
of order ``p`` or over the integers.
.. SEEALSO::
:class:`~sage.groups.matrix_gps.heisenberg.HeisenbergGroup`
EXAMPLES::
sage: H = groups.presentation.Heisenberg(); H
Finitely presented group < x1, y1, z |
x1*y1*x1^-1*y1^-1*z^-1, z*x1*z^-1*x1^-1, z*y1*z^-1*y1^-1 >
sage: H.order()
+Infinity
sage: r1, r2, r3 = H.relations()
sage: A = matrix([[1, 1, 0], [0, 1, 0], [0, 0, 1]])
sage: B = matrix([[1, 0, 0], [0, 1, 1], [0, 0, 1]])
sage: C = matrix([[1, 0, 1], [0, 1, 0], [0, 0, 1]])
sage: r1(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: r2(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: r3(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: p = 3
sage: Hp = groups.presentation.Heisenberg(p=3)
sage: Hp.order() == p**3
True
sage: Hnp = groups.presentation.Heisenberg(n=2, p=3)
sage: len(Hnp.relations())
13
REFERENCES:
- :wikipedia:`Heisenberg_group`
"""
n = Integer(n)
if n < 1:
raise ValueError('n must be a positive integer')
# generators' names are x1, .., xn, y1, .., yn, z
vx = ['x' + str(i) for i in range(1, n + 1)]
vy = ['y' + str(i) for i in range(1, n + 1)]
str_generators = ', '.join(vx + vy + ['z'])
F = FreeGroup(str_generators)
x = F.gens()[0:n] # list of generators x1, x2, ..., xn
y = F.gens()[n:2 * n] # list of generators x1, x2, ..., xn
z = F.gen(n * 2)
def commutator(a, b):
return a * b * a**-1 * b**-1
# First set of relations: [xi, yi] = z
r1 = [commutator(x[i], y[i]) * z**-1 for i in range(n)]
# Second set of relations: [z, xi] = 1
r2 = [commutator(z, x[i]) for i in range(n)]
# Third set of relations: [z, yi] = 1
r3 = [commutator(z, y[i]) for i in range(n)]
# Fourth set of relations: [xi, yi] = 1 for i != j
r4 = [commutator(x[i], y[j]) for i in range(n) for j in range(n) if i != j]
rls = r1 + r2 + r3 + r4
from sage.sets.primes import Primes
if p not in Primes() and p != 0:
raise ValueError("p must be 0 or a prime number")
if p > 0:
rls += [w**p for w in F.gens()]
return FinitelyPresentedGroup(F, tuple(rls))
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 FinitelyGeneratedHeisenbergPresentation(n=1, p=0):
r"""
Return a finite presentation of the Heisenberg group.
The Heisenberg group is the group of `(n+2) \times (n+2)` matrices
over a ring `R` with diagonal elements equal to 1, first row and
last column possibly nonzero, and all the other entries equal to zero.
INPUT:
- ``n`` -- the degree of the Heisenberg group
- ``p`` -- (optional) a prime number, where we construct the
Heisenberg group over the finite field `\ZZ/p\ZZ`
OUTPUT:
Finitely generated Heisenberg group over the finite field
of order ``p`` or over the integers.
.. SEEALSO::
:class:`~sage.groups.matrix_gps.heisenberg.HeisenbergGroup`
EXAMPLES::
sage: H = groups.presentation.Heisenberg(); H
Finitely presented group < x1, y1, z |
x1*y1*x1^-1*y1^-1*z^-1, z*x1*z^-1*x1^-1, z*y1*z^-1*y1^-1 >
sage: H.order()
+Infinity
sage: r1, r2, r3 = H.relations()
sage: A = matrix([[1, 1, 0], [0, 1, 0], [0, 0, 1]])
sage: B = matrix([[1, 0, 0], [0, 1, 1], [0, 0, 1]])
sage: C = matrix([[1, 0, 1], [0, 1, 0], [0, 0, 1]])
sage: r1(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: r2(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: r3(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: p = 3
sage: Hp = groups.presentation.Heisenberg(p=3)
sage: Hp.order() == p**3
True
sage: Hnp = groups.presentation.Heisenberg(n=2, p=3)
sage: len(Hnp.relations())
13
REFERENCES:
- :wikipedia:`Heisenberg_group`
"""
n = Integer(n)
if n < 1:
raise ValueError('n must be a positive integer')
# generators' names are x1, .., xn, y1, .., yn, z
vx = ['x' + str(i) for i in range(1,n+1)]
vy = ['y' + str(i) for i in range(1,n+1)]
str_generators = ', '.join(vx + vy + ['z'])
F = FreeGroup(str_generators)
x = F.gens()[0:n] # list of generators x1, x2, ..., xn
y = F.gens()[n:2*n] # list of generators x1, x2, ..., xn
z = F.gen(n*2)
def commutator(a, b): return a * b * a**-1 * b**-1
# First set of relations: [xi, yi] = z
r1 = [commutator(x[i], y[i]) * z**-1 for i in range(n)]
# Second set of relations: [z, xi] = 1
r2 = [commutator(z, x[i]) for i in range(n)]
# Third set of relations: [z, yi] = 1
r3 = [commutator(z, y[i]) for i in range(n)]
# Fourth set of relations: [xi, yi] = 1 for i != j
r4 = [commutator(x[i], y[j]) for i in range(n) for j in range(n) if i!=j]
rls = r1 + r2 + r3 + r4
from sage.sets.primes import Primes
if p not in Primes() and p != 0:
raise ValueError("p must be 0 or a prime number")
if p > 0:
rls += [w**p for w in F.gens()]
return FinitelyPresentedGroup(F, tuple(rls))
def FinitelyGeneratedAbelianPresentation(int_list):
r"""
Return canonical presentation of finitely generated abelian group.
INPUT:
- ``int_list`` -- List of integers defining the group to be returned, the defining list
is reduced to the invariants of the input list before generating the corresponding
group.
OUTPUT:
Finitely generated abelian group, `\ZZ_{n_1} \times \ZZ_{n_2} \times \cdots \times \ZZ_{n_k}`
as a finite presentation, where `n_i` forms the invariants of the input list.
EXAMPLES::
sage: groups.presentation.FGAbelian([2,2])
Finitely presented group < a, b | a^2, b^2, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([2,3])
Finitely presented group < a | a^6 >
sage: groups.presentation.FGAbelian([2,4])
Finitely presented group < a, b | a^2, b^4, a^-1*b^-1*a*b >
You can create free abelian groups::
sage: groups.presentation.FGAbelian([0])
Finitely presented group < a | >
sage: groups.presentation.FGAbelian([0,0])
Finitely presented group < a, b | a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([0,0,0])
Finitely presented group < a, b, c | a^-1*b^-1*a*b, a^-1*c^-1*a*c, b^-1*c^-1*b*c >
And various infinite abelian groups::
sage: groups.presentation.FGAbelian([0,2])
Finitely presented group < a, b | a^2, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([0,2,2])
Finitely presented group < a, b, c | a^2, b^2, a^-1*b^-1*a*b, a^-1*c^-1*a*c, b^-1*c^-1*b*c >
Outputs are reduced to minimal generators and relations::
sage: groups.presentation.FGAbelian([3,5,2,7,3])
Finitely presented group < a, b | a^3, b^210, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([3,210])
Finitely presented group < a, b | a^3, b^210, a^-1*b^-1*a*b >
The trivial group is an acceptable output::
sage: groups.presentation.FGAbelian([])
Finitely presented group < | >
sage: groups.presentation.FGAbelian([1])
Finitely presented group < | >
sage: groups.presentation.FGAbelian([1,1,1,1,1,1,1,1,1,1])
Finitely presented group < | >
Input list must consist of positive integers::
sage: groups.presentation.FGAbelian([2,6,3,9,-4])
Traceback (most recent call last):
...
ValueError: input list must contain nonnegative entries
sage: groups.presentation.FGAbelian([2,'a',4])
Traceback (most recent call last):
...
TypeError: unable to convert 'a' to an integer
TESTS::
sage: ag = groups.presentation.FGAbelian([2,2])
sage: ag.as_permutation_group().is_isomorphic(groups.permutation.KleinFour())
True
sage: G = groups.presentation.FGAbelian([2,4,8])
sage: C2 = CyclicPermutationGroup(2)
sage: C4 = CyclicPermutationGroup(4)
sage: C8 = CyclicPermutationGroup(8)
sage: gg = (C2.direct_product(C4)[0]).direct_product(C8)[0]
sage: gg.is_isomorphic(G.as_permutation_group())
True
sage: all(groups.presentation.FGAbelian([i]).as_permutation_group().is_isomorphic(groups.presentation.Cyclic(i).as_permutation_group()) for i in [2..35])
True
"""
from sage.groups.free_group import _lexi_gen
check_ls = [Integer(x) for x in int_list if Integer(x) >= 0]
if len(check_ls) != len(int_list):
raise ValueError('input list must contain nonnegative entries')
col_sp = diagonal_matrix(int_list).column_space()
invariants = FGP_Module(ZZ**(len(int_list)), col_sp).invariants()
name_gen = _lexi_gen()
F = FreeGroup([next(name_gen) for i in invariants])
ret_rls = [F([i+1])**invariants[i] for i in range(len(invariants)) if invariants[i]!=0]
# Build commutator relations
gen_pairs = [[F.gen(i),F.gen(j)] for i in range(F.ngens()-1) for j in range(i+1,F.ngens())]
ret_rls = ret_rls + [x[0]**(-1)*x[1]**(-1)*x[0]*x[1] for x in gen_pairs]
return FinitelyPresentedGroup(F, tuple(ret_rls))
def fundamental_group(f, simplified=True, projective=False):
r"""
Return a presentation of the fundamental group of the complement of
the algebraic set defined by the polynomial ``f``.
INPUT:
- ``f`` -- a polynomial in two variables, with coefficients in either
the rationals or a number field with a fixed embedding in `\QQbar`
- ``simplified`` -- boolean (default: ``True``); if set to ``True`` the
presentation will be simplified (see below)
- ``projective`` -- boolean (default: ``False``); if set to ``True``,
the fundamental group of the complement of the projective completion
of the curve will be computed, otherwise, the fundamental group of
the complement in the affine plane will be computed
If ``simplified`` is ``False``, the returned presentation has as
many generators as degree of the polynomial times the points in the
base used to create the segments that surround the discriminant. In
this case, the generators are granted to be meridians of the curve.
OUTPUT:
A presentation of the fundamental group of the complement of the
curve defined by ``f``.
EXAMPLES::
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
sage: R.<x,y> = QQ[]
sage: f = x^2 + y^3
sage: fundamental_group(f) # optional - sirocco
Finitely presented group < ... >
sage: fundamental_group(f, simplified=False) # optional - sirocco
Finitely presented group < ... >
::
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
sage: R.<x,y> = QQ[]
sage: f = y^3 + x^3
sage: fundamental_group(f) # optional - sirocco
Finitely presented group < ... >
It is also possible to have coefficients in a number field with a
fixed embedding in `\QQbar`::
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
sage: zeta = QQbar['x']('x^2+x+1').roots(multiplicities=False)[0]
sage: zeta
-0.50000000000000000? - 0.866025403784439?*I
sage: F = NumberField(zeta.minpoly(), 'zeta', embedding=zeta)
sage: F.inject_variables()
Defining zeta
sage: R.<x,y> = F[]
sage: f = y^3 + x^3 +zeta *x + 1
sage: fundamental_group(f) # optional - sirocco
Finitely presented group < x0 | >
"""
(x, y) = f.variables()
F = f.base_ring()
g = f.factor().radical().prod()
d = g.degree(y)
while not g.coefficient(y**d) in F or (projective and g.total_degree() > d):
g = g.subs({x: x + y})
d = g.degree(y)
disc = discrim(g)
segs = segments(disc)
vertices = list(set(flatten(segs)))
Faux = FreeGroup(d)
F = FreeGroup(d * len(vertices))
rels = []
if projective:
rels.append(prod(F.gen(i) for i in range(d)))
braidscomputed = braid_in_segment([(g, seg[0], seg[1]) for seg in segs])
for braidcomputed in braidscomputed:
seg = (braidcomputed[0][0][1], braidcomputed[0][0][2])
b = braidcomputed[1]
i = vertices.index(seg[0])
j = vertices.index(seg[1])
for k in range(d):
el1 = Faux([k + 1]) * b.inverse()
el2 = k + 1
w1 = F([sign(a)*d*i + a for a in el1.Tietze()])
w2 = F([d*j + el2])
rels.append(w1/w2)
G = F / rels
if simplified:
return G.simplified()
else:
return G
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)
