Coverage for local/lib/python2.7/site-packages/sage/homology/homology_group.py : 81%

""" 

Homology Groups 

This module defines a :meth:`HomologyGroup` class which is an abelian 

group that prints itself in a way that is suitable for homology 

groups. 

""" 

######################################################################## 

# Copyright (C) 2013 John H. Palmieri <palmieri@math.washington.edu> 

# Volker Braun <vbraun.name@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# 

# http://www.gnu.org/licenses/ 

######################################################################## 

from six.moves import range 

from sage.modules.free_module import VectorSpace 

from sage.groups.additive_abelian.additive_abelian_group import AdditiveAbelianGroup_fixed_gens 

from sage.rings.integer_ring import ZZ 

class HomologyGroup_class(AdditiveAbelianGroup_fixed_gens): 

""" 

Discrete Abelian group on `n` generators. This class inherits from 

:class:`~sage.groups.additive_abelian.additive_abelian_group.AdditiveAbelianGroup_fixed_gens`; 

see :mod:`sage.groups.additive_abelian.additive_abelian_group` for more 

documentation. The main difference between the classes is in the print 

representation. 

EXAMPLES:: 

sage: from sage.homology.homology_group import HomologyGroup 

sage: G = AbelianGroup(5, [5,5,7,8,9]); G 

Multiplicative Abelian group isomorphic to C5 x C5 x C7 x C8 x C9 

sage: H = HomologyGroup(5, ZZ, [5,5,7,8,9]); H 

C5 x C5 x C7 x C8 x C9 

sage: G == loads(dumps(G)) 

True 

sage: AbelianGroup(4) 

Multiplicative Abelian group isomorphic to Z x Z x Z x Z 

sage: HomologyGroup(4, ZZ) 

Z x Z x Z x Z 

sage: HomologyGroup(100, ZZ) 

Z^100 

""" 

def __init__(self, n, invfac): 

""" 

See :func:`HomologyGroup` for full documentation. 

EXAMPLES:: 

sage: from sage.homology.homology_group import HomologyGroup 

sage: H = HomologyGroup(5, ZZ, [5,5,7,8,9]); H 

C5 x C5 x C7 x C8 x C9 

""" 

n = len(invfac) 

A = ZZ ** n 

B = A.span([A.gen(i) * invfac[i] for i in range(n)]) 

AdditiveAbelianGroup_fixed_gens.__init__(self, A, B, A.gens()) 

self._original_invts = invfac 

def _repr_(self): 

""" 

Print representation of ``self``. 

EXAMPLES:: 

sage: from sage.homology.homology_group import HomologyGroup 

sage: H = HomologyGroup(7, ZZ, [4,4,4,4,4,7,7]) 

sage: H._repr_() 

'C4^5 x C7 x C7' 

sage: HomologyGroup(6, ZZ) 

Z^6 

""" 

eldv = self._original_invts 

if len(eldv) == 0: 

return "0" 

rank = len([x for x in eldv if x == 0]) 

torsion = sorted(x for x in eldv if x) 

if rank > 4: 

g = ["Z^%s" % rank] 

else: 

g = ["Z"] * rank 

if len(torsion) != 0: 

printed = [] 

for t in torsion: 

numfac = torsion.count(t) 

too_many = (numfac > 4) 

if too_many: 

if t not in printed: 

g.append("C{}^{}".format(t, numfac)) 

printed.append(t) 

else: 

g.append("C%s" % t) 

times = " x " 

return times.join(g) 

def _latex_(self): 

""" 

LaTeX representation of ``self``. 

EXAMPLES:: 

sage: from sage.homology.homology_group import HomologyGroup 

sage: H = HomologyGroup(7, ZZ, [4,4,4,4,4,7,7]) 

sage: H._latex_() 

'C_{4}^{5} \\times C_{7} \\times C_{7}' 

sage: latex(HomologyGroup(6, ZZ)) 

\ZZ^{6} 

""" 

eldv = self._original_invts 

if len(eldv) == 0: 

return "0" 

rank = len([x for x in eldv if x == 0]) 

torsion = sorted(x for x in eldv if x) 

if rank > 4: 

g = ["\\ZZ^{{{}}}".format(rank)] 

else: 

g = ["\\ZZ"] * rank 

if len(torsion) != 0: 

printed = [] 

for t in torsion: 

numfac = torsion.count(t) 

too_many = (numfac > 4) 

if too_many: 

if t not in printed: 

g.append("C_{{{}}}^{{{}}}".format(t, numfac)) 

printed.append(t) 

else: 

g.append("C_{{{}}}".format(t)) 

times = " \\times " 

return times.join(g) 

def HomologyGroup(n, base_ring, invfac=None): 

""" 

Abelian group on `n` generators which represents a homology group in a 

fixed degree. 

INPUT: 

- ``n`` -- integer; the number of generators 

- ``base_ring`` -- ring; the base ring over which the homology is computed 

- ``inv_fac`` -- list of integers; the invariant factors -- ignored 

if the base ring is a field 

OUTPUT: 

A class that can represent the homology group in a fixed 

homological degree. 

EXAMPLES:: 

sage: from sage.homology.homology_group import HomologyGroup 

sage: G = AbelianGroup(5, [5,5,7,8,9]); G 

Multiplicative Abelian group isomorphic to C5 x C5 x C7 x C8 x C9 

sage: H = HomologyGroup(5, ZZ, [5,5,7,8,9]); H 

C5 x C5 x C7 x C8 x C9 

sage: AbelianGroup(4) 

Multiplicative Abelian group isomorphic to Z x Z x Z x Z 

sage: HomologyGroup(4, ZZ) 

Z x Z x Z x Z 

sage: HomologyGroup(100, ZZ) 

Z^100 

""" 

if base_ring.is_field(): 

return VectorSpace(base_ring, n) 

# copied from AbelianGroup: 

if invfac is None: 

if isinstance(n, (list, tuple)): 

invfac = n 

n = len(n) 

else: 

invfac = [] 

if len(invfac) < n: 

invfac = [0] * (n - len(invfac)) + invfac 

elif len(invfac) > n: 

raise ValueError("invfac (={}) must have length n (={})".format(invfac, n)) 

M = HomologyGroup_class(n, invfac) 

return M