r"""
Absolute and relative (simplicial) cohomology of surfaces.
EXAMPLES:
The absolute cohomology of the regular octagon::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: S = translation_surfaces.regular_octagon()
sage: H = SimplicialCohomology(S)
A basis of cohomology::
sage: H.gens()
[{B[(0, 1)]: 1}, {B[(0, 2)]: 1}, {B[(0, 3)]: 1}, {B[(0, 0)]: 1}]
The absolute cohomology of the unfolding of the (3, 4, 13) triangle::
sage: from flatsurf import Polygon, similarity_surfaces
sage: P = Polygon(angles=[3, 4, 13])
sage: S = similarity_surfaces.billiard(P).minimal_cover(cover_type="translation")
sage: H = SimplicialCohomology(S)
sage: len(H.gens())
16
The relative cohomology, relative to the vertices::
sage: S = S.erase_marked_points() # optional: pyflatsurf # random output due to deprecation warnings
sage: H = SimplicialCohomology(S, relative=S.vertices()) # optional: pyflatsurf
sage: len(H.gens()) # optional: pyflatsurf
17
"""
######################################################################
# This file is part of sage-flatsurf.
#
# Copyright (C) 2022-2024 Julian Rüth
#
# sage-flatsurf is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#
# sage-flatsurf is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with sage-flatsurf. If not, see <https://www.gnu.org/licenses/>.
######################################################################
from sage.structure.parent import Parent
from sage.structure.element import Element
from sage.misc.cachefunc import cached_method
[docs]
class SimplicialCohomologyClass(Element):
r"""
A cohomology class.
INPUT:
- ``parent`` -- the cohomology group
- ``values`` -- a dict; the value at each generator of
:meth:`SimplicialCohomologyGroup.homology`.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: S = translation_surfaces.regular_octagon()
sage: H = SimplicialCohomology(S)
sage: f, _, _, _ = H.gens()
sage: from flatsurf.geometry.cohomology import SimplicialCohomologyClass
sage: isinstance(f, SimplicialCohomologyClass)
True
"""
def __init__(self, parent, values):
super().__init__(parent)
self._values = values
def _repr_(self):
r"""
Return a printable representation of this cohomology class.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: S = translation_surfaces.regular_octagon()
sage: H = SimplicialCohomology(S)
sage: f, _, _, _ = H.gens()
sage: f
{B[(0, 1)]: 1}
"""
return repr(self._values)
def __call__(self, homology):
r"""
Evaluate this class at an element of homology.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: H = SimplicialCohomology(T)
sage: γ = H.homology().gens()[0]
sage: f = H({γ: 1.337})
sage: f(γ)
1.33700000000000
"""
return sum(
self._values.get(gen, 0) * homology.coefficient(gen)
for gen in self.parent().homology().gens()
)
def _add_(self, other):
r"""
Return the pointwise sum of two cohomology classes.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: H = SimplicialCohomology(T)
sage: γ = H.homology().gens()[0]
sage: f = H({γ: 1.337})
sage: (f + f)(γ) == 2*f(γ)
True
"""
other = self.parent()(other)
values = {}
for gen in self.parent().homology().gens():
value = self(gen) + other(gen)
if value:
values[gen] = value
return self.parent()(values)
def _richcmp_(self, other, op):
r"""
Return how this cohomoly class compares to other with respect to the
binary relation ``op``.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: S = translation_surfaces.regular_octagon()
sage: H = SimplicialCohomology(S)
sage: f, g, _, _ = H.gens()
sage: f == g
False
"""
from sage.structure.richcmp import op_EQ, op_NE
if op == op_NE:
return not self._richcmp_(other, op_EQ)
if op == op_EQ:
if self is other:
return True
if self.parent() != other.parent():
return False
return self._values == other._values
return super()._richcmp_(other, op)
[docs]
class SimplicialCohomologyGroup(Parent):
r"""
The ``k``-th simplicial cohomology group of the ``surface`` with
``coefficients``.
INPUT:
- ``surface`` -- a finite type surface without boundary
- ``k`` -- an integer
- ``coefficients`` -- a ring
- ``relative`` -- a subset of points of the ``surface``
- ``implementation`` -- a string; the algorithm used to compute the
cohomology, only ``"dual`` is supported at the moment.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: SimplicialCohomology(T)
H¹(Translation Surface in H_1(0) built from a square)
"""
Element = SimplicialCohomologyClass
def __init__(self, surface, k, coefficients, relative, implementation, category):
Parent.__init__(self, category=category)
if surface.is_mutable():
raise TypeError("surface most be immutable")
from sage.all import ZZ
if k not in ZZ:
raise TypeError("k must be an integer")
from sage.categories.all import Rings
if coefficients not in Rings():
raise TypeError("coefficients must be a ring")
if implementation == "dual":
if not surface.is_compact():
raise NotImplementedError(
"dual implementation can only handle cohomology of compact surfaces"
)
if surface.is_with_boundary():
raise NotImplementedError(
"dual implementation can only handle cohomology of surfaces without boundary"
)
if coefficients.characteristic() > 0:
raise NotImplementedError(
"dual implementation can only handle cohomology with coefficients of characteristic zero"
)
if not coefficients.is_integral_domain():
raise NotImplementedError(
"dual implementation can only handle cohomology with flat coefficient rings"
)
else:
raise NotImplementedError("unknown implementation for cohomology group")
self._surface = surface
self._k = k
self._coefficients = coefficients
self._relative = relative
[docs]
def is_absolute(self):
r"""
Return whether this is an absolute cohomology (and not a relative one).
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: H = SimplicialCohomology(T)
sage: H.is_absolute()
True
"""
return not self._relative
def _repr_(self):
r"""
Return a printable representation of this cohomology group.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: SimplicialCohomology(T)
H¹(Translation Surface in H_1(0) built from a square)
sage: SimplicialCohomology(T, relative=T.vertices())
H¹(Translation Surface in H_1(0) built from a square, {Vertex 0 of polygon 0})
sage: SimplicialCohomology(T, coefficients=CC, relative=T.vertices())
H¹(Translation Surface in H_1(0) built from a square, {Vertex 0 of polygon 0}; Complex Field with 53 bits of precision)
"""
if self._k == 0:
k = "⁰"
elif self._k == 1:
k = "¹"
elif self._k == 2:
k = "²"
else:
k = f"^{self._k}"
Hk = f"H{k}"
X = repr(self.surface())
if not self.is_absolute():
X = f"{X}, {set(self._relative)}"
from sage.all import RR
if self._coefficients is not RR:
sep = ";"
X = f"{X}{sep} {self._coefficients}"
return f"{Hk}({X})"
[docs]
def surface(self):
r"""
Return the surface over which this cohomology is defined.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: H = SimplicialCohomology(T)
sage: H.surface() is T
True
"""
return self._surface
def _element_constructor_(self, x):
r"""
Return ``x`` as a class in this cohomology.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: H = SimplicialCohomology(T)
sage: H(0)
{}
sage: H({gen: 1 for gen in H.homology().gens()})
{B[(0, 1)]: 1, B[(0, 0)]: 1}
"""
if not x:
x = {}
if isinstance(x, dict):
x = {self.homology()(gen): value for (gen, value) in x.items() if value}
return self.element_class(self, x)
raise NotImplementedError("cannot create a cohomology class from this data")
[docs]
@cached_method
def homology(self):
r"""
Return the homology of the underlying space (with integer
coefficients).
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: H = SimplicialCohomology(T)
sage: H.homology()
H₁(Translation Surface in H_1(0) built from a square)
"""
from flatsurf.geometry.homology import SimplicialHomology
return SimplicialHomology(self._surface, self._k, relative=self._relative)
[docs]
def gens(self):
r"""
Return generators of this cohomology.
EXAMPLES::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: H = SimplicialCohomology(T)
sage: H.gens()
[{B[(0, 1)]: 1}, {B[(0, 0)]: 1}]
"""
return [self({gen: 1}) for gen in self.homology().gens()]
[docs]
def SimplicialCohomology(
surface, k=1, coefficients=None, relative=None, implementation="dual", category=None
):
r"""
Return the ``k``-th simplicial cohomology group of ``surface``.
INPUT:
- ``surface`` -- a surface
- ``k`` -- an integer (default: ``1``)
- ``coefficients`` -- a ring (default: the reals); consider cohomology with
coefficients in this ring
- ``relative`` -- a set (default: the empty set); if non-empty, then
relative cohomology with respect to this set is constructed.
- ``implementation`` -- a string (default: ``"dual"``); the algorithm used
to compute the cohomology groups. Currently only ``"dual"`` is supported,
i.e., the groups are computed as duals of the generic homology groups
from SageMath.
- ``category`` -- a category; if not specified, a category for the
cohomology group is chosen automatically depending on ``coefficients``.
TESTS:
Cohomology is unique and cached::
sage: from flatsurf import translation_surfaces, SimplicialCohomology
sage: T = translation_surfaces.square_torus()
sage: SimplicialCohomology(T) is SimplicialCohomology(T)
True
"""
return surface.cohomology(k, coefficients, relative, implementation, category)