Split chains/cochains as separate parents and expose cup product

src/sage/homology/cell_complex.py

@@ -33,14 +33,26 @@
     :meth:`~GenericCellComplex.homology` method for cell complexes --
     just make sure it gets documented.
 """
-from __future__ import absolute_import
+
+########################################################################
+#       Copyright (C) 2009 John H. Palmieri <palmieri@math.washington.edu>
+#
+#  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 __future__ import absolute_import
+
 from sage.structure.sage_object import SageObject
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
-from sage.combinat.free_module import CombinatorialFreeModule, CombinatorialFreeModuleElement
 from sage.misc.abstract_method import abstract_method
+from sage.homology.chains import Chains, Cochains
+
 
 class GenericCellComplex(SageObject):
     r"""
@@ -744,7 +756,11 @@ def n_chains(self, n, base_ring=ZZ, cochains=False):
             sage: list(simplicial_complexes.Sphere(2).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)]
         """
-        return Chains(tuple(self.n_cells(n)), base_ring, cochains)
+        n_cells = tuple(self.n_cells(n))
+        if cochains:
+            return Cochains(self, n, n_cells, base_ring)
+        else:
+            return Chains(self, n, n_cells, base_ring)
 
     def algebraic_topological_model(self, base_ring=QQ):
         r"""
@@ -1115,131 +1131,3 @@ def _repr_(self):
         return Name + " complex " + vertex_string + cells_string
 
 
-class Chains(CombinatorialFreeModule):
-    r"""
-    Class for the free module of chains and/or cochains in a given
-    degree.
-
-    INPUT:
-
-    - ``n_cells`` -- tuple of `n`-cells, which thus forms a basis for
-      this module
-    - ``base_ring`` -- optional (default `\ZZ`)
-    - ``cochains`` -- boolean (optional, default ``False``); if
-      ``True``, return cochains instead
-
-    One difference between chains and cochains is notation. In a
-    simplicial complex, for example, a simplex ``(0,1,2)`` is written
-    as "(0,1,2)" in the group of chains but as "\chi_(0,1,2)" in the
-    group of cochains.
-
-    Also, since the free modules of chains and cochains are dual,
-    there is a pairing `\langle c, z \rangle`, sending a cochain `c`
-    and a chain `z` to a scalar.
-
-    EXAMPLES::
-
-        sage: S2 = simplicial_complexes.Sphere(2)
-        sage: C_2 = S2.n_chains(1)
-        sage: C_2_co = S2.n_chains(1, cochains=True)
-        sage: x = C_2.basis()[Simplex((0,2))]
-        sage: y = C_2.basis()[Simplex((1,3))]
-        sage: z = x+2*y
-        sage: a = C_2_co.basis()[Simplex((1,3))]
-        sage: b = C_2_co.basis()[Simplex((0,3))]
-        sage: c = 3*a-2*b
-        sage: z
-        (0, 2) + 2*(1, 3)
-        sage: c
-        -2*\chi_(0, 3) + 3*\chi_(1, 3)
-        sage: c.eval(z)
-        6
-    """
-    def __init__(self, n_cells, base_ring=None, cochains=False):
-        """
-        EXAMPLES::
-
-            sage: T = cubical_complexes.Torus()
-            sage: T.n_chains(2, QQ)
-            Free module generated by {[1,1] x [0,1] x [1,1] x [0,1],
-             [0,0] x [0,1] x [0,1] x [1,1], [0,0] x [0,1] x [1,1] x [0,1],
-             [0,0] x [0,1] x [0,0] x [0,1], [0,1] x [1,1] x [0,1] x [0,0],
-             [0,1] x [0,0] x [0,0] x [0,1], [1,1] x [0,1] x [0,1] x [0,0],
-             [0,1] x [1,1] x [0,0] x [0,1], [0,0] x [0,1] x [0,1] x [0,0],
-             [0,1] x [0,0] x [0,1] x [0,0], [0,1] x [0,0] x [1,1] x [0,1],
-             [0,1] x [1,1] x [1,1] x [0,1], [0,1] x [0,0] x [0,1] x [1,1],
-             [1,1] x [0,1] x [0,0] x [0,1], [1,1] x [0,1] x [0,1] x [1,1],
-             [0,1] x [1,1] x [0,1] x [1,1]} over Rational Field
-            sage: T.n_chains(2).dimension()
-            16
-
-        TESTS::
-
-            sage: T.n_chains(2).base_ring()
-            Integer Ring
-            sage: T.n_chains(8).dimension()
-            0
-            sage: T.n_chains(-3).dimension()
-            0
-        """
-        if base_ring is None:
-            base_ring=ZZ
-        self._cochains = cochains
-        if cochains:
-            CombinatorialFreeModule.__init__(self, base_ring, n_cells,
-                                             prefix='\\chi', bracket=['_', ''])
-        else:
-            CombinatorialFreeModule.__init__(self, base_ring, n_cells,
-                                             prefix='', bracket=False)
-
-    class Element(CombinatorialFreeModuleElement):
-
-        def eval(self, other):
-            """
-            Evaluate this cochain on the chain ``other``.
-
-            INPUT:
-
-            - ``other`` -- a chain for the same cell complex in the
-              same dimension with the same base ring
-
-            OUTPUT: scalar
-
-            EXAMPLES::
-
-                sage: S2 = simplicial_complexes.Sphere(2)
-                sage: C_2 = S2.n_chains(1)
-                sage: C_2_co = S2.n_chains(1, cochains=True)
-                sage: x = C_2.basis()[Simplex((0,2))]
-                sage: y = C_2.basis()[Simplex((1,3))]
-                sage: z = x+2*y
-                sage: a = C_2_co.basis()[Simplex((1,3))]
-                sage: b = C_2_co.basis()[Simplex((0,3))]
-                sage: c = 3*a-2*b
-                sage: z
-                (0, 2) + 2*(1, 3)
-                sage: c
-                -2*\chi_(0, 3) + 3*\chi_(1, 3)
-                sage: c.eval(z)
-                6
-
-            TESTS::
-
-                sage: z.eval(c) # z is not a cochain
-                Traceback (most recent call last):
-                ...
-                ValueError: this element is not a cochain
-                sage: c.eval(c) # can't evaluate a cochain on a cochain
-                Traceback (most recent call last):
-                ...
-                ValueError: the elements are not compatible
-            """
-            if not self.parent()._cochains:
-                raise ValueError('this element is not a cochain')
-            if not (other.parent().indices() == self.parent().indices()
-                    and other.base_ring() == self.base_ring()
-                    and not other.parent()._cochains):
-                raise ValueError('the elements are not compatible')
-            result = sum(coeff * other.coefficient(cell) for cell, coeff in self)
-            return result
-