gh-41230: some typing in homology
    
just adding a few basic typing annotations in the homology folder

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #41230
Reported by: Frédéric Chapoton
Reviewer(s): Chenxin Zhong

Author: Release Manager

src/sage/homology/chain_complex.py

@@ -317,7 +317,7 @@ def ChainComplex(data=None, base_ring=None, grading_group=None,
 
 class Chain_class(ModuleElement):
 
-    def __init__(self, parent, vectors, check=True):
+    def __init__(self, parent, vectors, check=True) -> None:
         r"""
         A Chain in a Chain Complex.
 
@@ -370,7 +370,7 @@ def vector(self, degree):
         except KeyError:
             return self.parent().free_module(degree).zero()
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Print representation.
 
@@ -510,7 +510,7 @@ def vector_art(d):
             concatenated += UnicodeArt([' ... ']) + r
         return concatenated
 
-    def is_cycle(self):
+    def is_cycle(self) -> bool:
         """
         Return whether the chain is a cycle.
 
@@ -531,7 +531,7 @@ def is_cycle(self):
                 return False
         return True
 
-    def is_boundary(self):
+    def is_boundary(self) -> bool:
         """
         Return whether the chain is a boundary.
 
@@ -607,7 +607,7 @@ def _lmul_(self, scalar):
         parent = self.parent()
         return parent.element_class(parent, vectors)
 
-    def __eq__(self, other):
+    def __eq__(self, other) -> bool:
         """
         Return ``True`` if this chain is equal to ``other``.
 
@@ -624,7 +624,7 @@ def __eq__(self, other):
             return False
         return self._vec == other._vec
 
-    def __ne__(self, other):
+    def __ne__(self, other) -> bool:
         """
         Return ``True`` if this chain is not equal to ``other``.
 
@@ -666,7 +666,8 @@ class ChainComplex_class(Parent):
         sage: D
         Chain complex with at most 2 nonzero terms over Integer Ring
     """
-    def __init__(self, grading_group, degree_of_differential, base_ring, differentials):
+    def __init__(self, grading_group, degree_of_differential, base_ring,
+                 differentials) -> None:
         """
         Initialize ``self``.
 
@@ -690,7 +691,8 @@ def __init__(self, grading_group, degree_of_differential, base_ring, differentia
         if any(dim+degree_of_differential not in differentials and d.nrows() != 0
                for dim, d in differentials.items()):
             raise ValueError('invalid differentials')
-        if any(dim-degree_of_differential not in differentials and d.ncols() != 0
+        if any(dim - degree_of_differential not in differentials
+               and d.ncols() != 0
                for dim, d in differentials.items()):
             raise ValueError('invalid differentials')
         self._grading_group = grading_group
@@ -1059,7 +1061,7 @@ def free_module(self, degree=None):
             rank = self.free_module_rank(degree)
         return FreeModule(self.base_ring(), rank)
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         The hash is formed by combining the hashes of.
 
@@ -1078,7 +1080,7 @@ def __hash__(self):
                 ^ hash(tuple(self.differential().items()))
                 ^ hash(self.degree_of_differential()))
 
-    def __eq__(self, other):
+    def __eq__(self, other) -> bool:
         """
         Return ``True`` iff this chain complex is the same as other: that
         is, if the base rings and the matrices of the two are the
@@ -1109,7 +1111,7 @@ def __eq__(self, other):
                 equal = equal and mat.ncols() == 0 and mat.nrows() == 0
         return equal
 
-    def __ne__(self, other):
+    def __ne__(self, other) -> bool:
         """
         Return ``True`` iff this chain complex is not the same as other.
 
@@ -1607,7 +1609,7 @@ def shift(self, n=1):
         return ChainComplex({k-shift: sgn * self._diff[k] for k in self._diff},
                             degree_of_differential=deg)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Print representation.
 
@@ -1622,7 +1624,7 @@ def _repr_(self):
             s = 'Trivial chain complex'
         else:
             s = 'Chain complex with at most {0} nonzero terms'.format(len(diffs)-1)
-        s += ' over {0}'.format(self.base_ring())
+        s += f' over {self.base_ring()}'
         return s
 
     def _ascii_art_(self):

src/sage/homology/chain_complex_morphism.py

@@ -95,7 +95,7 @@ class ChainComplexMorphism(Morphism):
     """
     An element of this class is a morphism of chain complexes.
     """
-    def __init__(self, matrices, C, D, check=True):
+    def __init__(self, matrices, C, D, check=True) -> None:
         """
         Create a morphism from a dictionary of matrices.
 
@@ -515,7 +515,7 @@ def __sub__(self, x):
         """
         return self + (-x)
 
-    def __eq__(self, x):
+    def __eq__(self, x) -> bool:
         """
         Return ``True`` if and only if ``self == x``.
 
@@ -581,7 +581,7 @@ def is_surjective(self) -> bool:
         m = self.to_matrix()
         return m.rank() == m.nrows()
 
-    def is_injective(self):
+    def is_injective(self) -> bool:
         """
         Return ``True`` if this map is injective.
 
@@ -604,7 +604,7 @@ def is_injective(self):
         """
         return self.to_matrix().right_nullity() == 0
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         TESTS::
 
@@ -616,7 +616,7 @@ def __hash__(self):
         """
         return hash(self.domain()) ^ hash(self.codomain()) ^ hash(tuple(self._matrix_dictionary.items()))
 
-    def _repr_type(self):
+    def _repr_type(self) -> str:
         """
         EXAMPLES::
 

src/sage/homology/chain_homotopy.py

@@ -109,7 +109,7 @@ class ChainHomotopy(Morphism):
         ...
         ValueError: the data do not define a valid chain homotopy
     """
-    def __init__(self, matrices, f, g=None):
+    def __init__(self, matrices, f, g=None) -> None:
         r"""
         Create a chain homotopy between the given chain maps
         from a dictionary of matrices.
@@ -190,7 +190,7 @@ def __init__(self, matrices, f, g=None):
         self._g = g
         Morphism.__init__(self, Hom(domain, codomain))
 
-    def is_algebraic_gradient_vector_field(self):
+    def is_algebraic_gradient_vector_field(self) -> bool:
         r"""
         An algebraic gradient vector field is a linear map
         `H: C \to C` such that `H H = 0`.
@@ -242,7 +242,7 @@ def is_algebraic_gradient_vector_field(self):
                     return False
         return True
 
-    def is_homology_gradient_vector_field(self):
+    def is_homology_gradient_vector_field(self) -> bool:
         r"""
         A homology gradient vector field is an algebraic gradient vector
         field `H: C \to C` (i.e., a chain homotopy satisfying `H
@@ -345,7 +345,7 @@ def dual(self):
         matrices = {i-deg: matrix_dict[i].transpose() for i in matrix_dict}
         return ChainHomotopy(matrices, self._f.dual(), self._g.dual())
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         TESTS::
 
@@ -359,7 +359,7 @@ def __hash__(self):
         """
         return hash(self._f) ^ hash(self._g) ^ hash(tuple(self._matrix_dictionary.items()))
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         String representation.
 
@@ -422,7 +422,7 @@ class ChainContraction(ChainHomotopy):
         ....:                       1: zero_matrix(ZZ, 1),
         ....:                       2: identity_matrix(ZZ, 1)}, pi, iota)
     """
-    def __init__(self, matrices, pi, iota):
+    def __init__(self, matrices, pi, iota) -> None:
         r"""
         Create a chain contraction from the given data.
 

src/sage/homology/chains.py

@@ -27,7 +27,7 @@
 
 class CellComplexReference:
 
-    def __init__(self, cell_complex, degree, cells=None):
+    def __init__(self, cell_complex, degree, cells=None) -> None:
         """
         Auxiliary base class for chains and cochains.
 
@@ -124,7 +124,8 @@ class Chains(CellComplexReference, CombinatorialFreeModule):
         sage: c.eval(z)
         6
     """
-    def __init__(self, cell_complex, degree, cells=None, base_ring=None):
+    def __init__(self, cell_complex, degree, cells=None,
+                 base_ring=None) -> None:
         """
         EXAMPLES::
 
@@ -264,7 +265,7 @@ def boundary(self):
             )
             return codomain.from_vector(d * self.to_vector())
 
-        def is_cycle(self):
+        def is_cycle(self) -> bool:
             """
             Test whether the chain is a cycle.
 
@@ -288,7 +289,7 @@ def is_cycle(self):
             """
             return self.to_complex().is_cycle()
 
-        def is_boundary(self):
+        def is_boundary(self) -> bool:
             """
             Test whether the chain is a boundary.
 
@@ -351,7 +352,8 @@ class Cochains(CellComplexReference, CombinatorialFreeModule):
         sage: c.eval(z)
         6
     """
-    def __init__(self, cell_complex, degree, cells=None, base_ring=None):
+    def __init__(self, cell_complex, degree, cells=None,
+                 base_ring=None) -> None:
         """
         EXAMPLES::
 
@@ -493,7 +495,7 @@ def coboundary(self):
             )
             return codomain.from_vector(d * self.to_vector())
 
-        def is_cocycle(self):
+        def is_cocycle(self) -> bool:
             """
             Test whether the cochain is a cocycle.
 
@@ -517,7 +519,7 @@ def is_cocycle(self):
             """
             return self.to_complex().is_cycle()
 
-        def is_coboundary(self):
+        def is_coboundary(self) -> bool:
             """
             Test whether the cochain is a coboundary.
 

src/sage/homology/free_resolution.py

@@ -180,7 +180,7 @@ def __classcall_private__(cls, module, *args, graded=False, degrees=None, shifts
             return module.graded_free_resolution(*args, **kwds)
         return module.free_resolution(*args, **kwds)
 
-    def __init__(self, module, name='S', **kwds):
+    def __init__(self, module, name='S', **kwds) -> None:
         """
         Initialize ``self``.
 
@@ -206,7 +206,7 @@ def __init__(self, module, name='S', **kwds):
         self._name = name
         self._module = module
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -224,7 +224,7 @@ def _repr_(self):
             return f"Free resolution of the row space of the matrix:\n{self._module}"
         return f"Free resolution of {self._module}"
 
-    def _repr_module(self, i):
+    def _repr_module(self, i) -> str:
         r"""
         Return the string form of the `i`-th free module.
 
@@ -369,7 +369,7 @@ def _length(self):
         """
         return len(self._maps)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string form of this resolution.
 
@@ -851,7 +851,7 @@ class FiniteFreeResolution_singular(FiniteFreeResolution):
         [-y*z + x*w]
         [ z^2 - y*w]
     """
-    def __init__(self, module, name='S', algorithm='heuristic', **kwds):
+    def __init__(self, module, name='S', algorithm='heuristic', **kwds) -> None:
         r"""
         Initialize ``self``.
 

src/sage/homology/hochschild_complex.py

@@ -83,7 +83,7 @@ class HochschildComplex(UniqueRepresentation, Parent):
     - https://ncatlab.org/nlab/show/Hochschild+cohomology
     - [Red2001]_
     """
-    def __init__(self, A, M):
+    def __init__(self, A, M) -> None:
         """
         Initialize ``self``.
 
@@ -105,7 +105,7 @@ def __init__(self, A, M):
         Parent.__init__(self, base=A.base_ring(),
                         category=ChainComplexes(A.base_ring()))
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return a string representation of ``self``.
 
@@ -121,7 +121,7 @@ def _repr_(self):
         """
         return "Hochschild complex of {} with coefficients in {}".format(self._A, self._M)
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a latex representation of ``self``.
 
@@ -570,7 +570,7 @@ class Element(ModuleElement):
             sage: H({0: x-y, 2: H.module(2).basis().an_element()})
             Chain with 2 nonzero terms over Integer Ring
         """
-        def __init__(self, parent, vectors):
+        def __init__(self, parent, vectors) -> None:
             """
             Initialize ``self``.
 
@@ -601,7 +601,7 @@ def vector(self, degree):
             except KeyError:
                 return self.parent().module(degree).zero()
 
-        def _repr_(self):
+        def _repr_(self) -> str:
             """
             Print representation.
 
@@ -725,7 +725,7 @@ def _lmul_(self, scalar):
                     vectors[d] = vec
             return self.__class__(self.parent(), vectors)
 
-        def _richcmp_(self, other, op):
+        def _richcmp_(self, other, op) -> bool:
             """
             Rich comparison of ``self`` to ``other``.
 

src/sage/homology/homology_group.py

@@ -45,7 +45,7 @@ class HomologyGroup_class(AdditiveAbelianGroup_fixed_gens):
         sage: HomologyGroup(100, ZZ)
         Z^100
     """
-    def __init__(self, n, invfac):
+    def __init__(self, n, invfac) -> None:
         """
         See :func:`HomologyGroup` for full documentation.
 
@@ -62,7 +62,7 @@ def __init__(self, n, invfac):
         AdditiveAbelianGroup_fixed_gens.__init__(self, A, B, A.gens())
         self._original_invts = invfac
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Print representation of ``self``.