diff --git a/src/sage/algebras/hecke_algebras/cubic_hecke_matrix_rep.py b/src/sage/algebras/hecke_algebras/cubic_hecke_matrix_rep.py
index 9b12b49eecb..a3311085ca5 100644
--- a/src/sage/algebras/hecke_algebras/cubic_hecke_matrix_rep.py
+++ b/src/sage/algebras/hecke_algebras/cubic_hecke_matrix_rep.py
@@ -74,7 +74,7 @@ class RepresentationType(Enum):
         sage: chmr.RepresentationType.RegularLeft.is_regular()
         True
     """
-    def is_split(self):
+    def is_split(self) -> bool:
         r"""
         Return ``True`` if this representation type is absolutely split,
         ``False`` else-wise.
@@ -88,7 +88,7 @@ def is_split(self):
         """
         return self.value['split']
 
-    def is_regular(self):
+    def is_regular(self) -> bool:
         r"""
         Return ``True`` if this representation type is regular, ``False``
         else-wise.
diff --git a/src/sage/categories/crystals.py b/src/sage/categories/crystals.py
index c32f99895a0..5c995a0a632 100644
--- a/src/sage/categories/crystals.py
+++ b/src/sage/categories/crystals.py
@@ -152,7 +152,7 @@ def example(self, choice='highwt', **kwds):
 
     class MorphismMethods:
         @cached_method
-        def is_isomorphism(self):
+        def is_isomorphism(self) -> bool:
             """
             Check if ``self`` is a crystal isomorphism.
 
@@ -178,7 +178,7 @@ def is_isomorphism(self):
 
         # TODO: This could be moved to sets
         @cached_method
-        def is_embedding(self):
+        def is_embedding(self) -> bool:
             """
             Check if ``self`` is an injective crystal morphism.
 
@@ -213,7 +213,7 @@ def is_embedding(self):
             return True
 
         @cached_method
-        def is_strict(self):
+        def is_strict(self) -> bool:
             """
             Check if ``self`` is a strict crystal morphism.
 
@@ -1301,7 +1301,7 @@ def number_of_connected_components(self):
             """
             return len(self.connected_components_generators())
 
-        def is_connected(self):
+        def is_connected(self) -> bool:
             """
             Return ``True`` if ``self`` is a connected crystal.
 
@@ -1528,7 +1528,7 @@ def s(self, i):
                     b = b.e(i)
             return b
 
-        def is_highest_weight(self, index_set=None):
+        def is_highest_weight(self, index_set=None) -> bool:
             r"""
             Return ``True`` if ``self`` is a highest weight.
 
@@ -1550,9 +1550,10 @@ def is_highest_weight(self, index_set=None):
                 index_set = self.index_set()
             return all(self.e(i) is None for i in index_set)
 
-        def is_lowest_weight(self, index_set=None):
+        def is_lowest_weight(self, index_set=None) -> bool:
             r"""
             Return ``True`` if ``self`` is a lowest weight.
+
             Specifying the option ``index_set`` to be a subset `I` of the
             index set of the underlying crystal, finds all lowest
             weight vectors for arrows in `I`.
diff --git a/src/sage/categories/monoids.py b/src/sage/categories/monoids.py
index 313ec5d6374..2e8ebae94a0 100644
--- a/src/sage/categories/monoids.py
+++ b/src/sage/categories/monoids.py
@@ -274,7 +274,7 @@ def _div_(left, right):
             """
             return left * ~right
 
-        def is_one(self):
+        def is_one(self) -> bool:
             r"""
             Return whether ``self`` is the one of the monoid.
 
@@ -603,7 +603,7 @@ def algebra_generators(self):
 
         class ElementMethods:
 
-            def is_central(self):
+            def is_central(self) -> bool:
                 r"""
                 Return whether the element ``self`` is central.
 
diff --git a/src/sage/combinat/partition_kleshchev.py b/src/sage/combinat/partition_kleshchev.py
index 444a46f2cf1..85a9a30bf39 100644
--- a/src/sage/combinat/partition_kleshchev.py
+++ b/src/sage/combinat/partition_kleshchev.py
@@ -428,7 +428,7 @@ def mullineux_conjugate(self):
         KP = mu.parent()
         return KP.element_class(KP, mu.add_cell(*mu.cogood_cells( r-c-self.parent()._multicharge[0]) ))
 
-    def is_regular(self):
+    def is_regular(self) -> bool:
         r"""
         Return ``True`` if ``self`` is a `e`-regular partition tuple.
 
@@ -454,7 +454,7 @@ def is_regular(self):
         KP = self.parent()
         return super().is_regular(KP._e, KP._multicharge)
 
-    def is_restricted(self):
+    def is_restricted(self) -> bool:
         r"""
         Return ``True`` if ``self`` is an `e`-restricted partition tuple.
 
@@ -811,7 +811,7 @@ def mullineux_conjugate(self):
         KP = mu.parent()
         return KP.element_class(KP, mu.add_cell(*mu.cogood_cells( r-c-self.parent()._multicharge[k])))
 
-    def is_regular(self):
+    def is_regular(self) -> bool:
         r"""
         Return ``True`` if ``self`` is a `e`-regular partition tuple.
 
@@ -835,7 +835,7 @@ def is_regular(self):
         KP = self.parent()
         return _is_regular(self.to_list(), KP._multicharge, KP._convention)
 
-    def is_restricted(self):
+    def is_restricted(self) -> bool:
         r"""
         Return ``True`` if ``self`` is an `e`-restricted partition tuple.
 
diff --git a/src/sage/combinat/root_system/reflection_group_complex.py b/src/sage/combinat/root_system/reflection_group_complex.py
index c213a2dcf6c..38c23595a44 100644
--- a/src/sage/combinat/root_system/reflection_group_complex.py
+++ b/src/sage/combinat/root_system/reflection_group_complex.py
@@ -852,7 +852,7 @@ def discriminant_in_invariant_ring(self, invariants=None):
         return sum(coeffs[i] * mons[i] for i in range(m))
 
     @cached_method
-    def is_crystallographic(self):
+    def is_crystallographic(self) -> bool:
         r"""
         Return ``True`` if ``self`` is crystallographic.
 
@@ -890,9 +890,10 @@ def is_crystallographic(self):
             sage: W.is_crystallographic()
             False
         """
-        return self.is_real() and all(t.to_matrix().base_ring() is QQ for t in self.simple_reflections())
+        return self.is_real() and all(t.to_matrix().base_ring() is QQ
+                                      for t in self.simple_reflections())
 
-    def number_of_irreducible_components(self):
+    def number_of_irreducible_components(self) -> int:
         r"""
         Return the number of irreducible components of ``self``.
 
@@ -908,7 +909,7 @@ def number_of_irreducible_components(self):
         """
         return len(self._type)
 
-    def irreducible_components(self):
+    def irreducible_components(self) -> list:
         r"""
         Return a list containing the irreducible components of ``self``
         as finite reflection groups.
diff --git a/src/sage/combinat/species/structure.py b/src/sage/combinat/species/structure.py
index 43dbd854ba7..87619524b28 100644
--- a/src/sage/combinat/species/structure.py
+++ b/src/sage/combinat/species/structure.py
@@ -182,11 +182,10 @@ def _relabel(self, i):
             [1, 2, 3]
         """
         if isinstance(i, (int, Integer)):
-            return self._labels[i-1]
-        else:
-            return i
+            return self._labels[i - 1]
+        return i
 
-    def is_isomorphic(self, x):
+    def is_isomorphic(self, x) -> bool:
         """
         EXAMPLES::
 
@@ -203,7 +202,7 @@ def is_isomorphic(self, x):
         if self.parent() != x.parent():
             return False
 
-        #We don't care about the labels for isomorphism testing
+        # We don't care about the labels for isomorphism testing
         return self.canonical_label()._list == x.canonical_label()._list
 
 
diff --git a/src/sage/combinat/subword_complex.py b/src/sage/combinat/subword_complex.py
index 8dcbd2d64cd..f2d8b46f7e7 100644
--- a/src/sage/combinat/subword_complex.py
+++ b/src/sage/combinat/subword_complex.py
@@ -349,7 +349,7 @@ def root_configuration(self):
         Phi = self.parent().group().roots()
         return [Phi[i] for i in self._root_configuration_indices()]
 
-    def kappa_preimage(self):
+    def kappa_preimage(self) -> list:
         r"""
         Return the fiber of ``self`` under the `\kappa` map.
 
@@ -403,7 +403,7 @@ def kappa_preimage(self):
                 if all(w.action_on_root_indices(i, side='left') < N
                        for i in root_conf)]
 
-    def is_vertex(self):
+    def is_vertex(self) -> bool:
         r"""
         Return ``True`` if ``self`` is a vertex of the brick polytope
         of ``self.parent``.
@@ -1445,7 +1445,7 @@ def greedy_facet(self, side='positive'):
 
     # topological properties
 
-    def is_sphere(self):
+    def is_sphere(self) -> bool:
         r"""
         Return ``True`` if the subword complex ``self`` is a sphere.
 
@@ -1472,7 +1472,7 @@ def is_sphere(self):
         w = W.demazure_product(self._Q)
         return w == self._pi
 
-    def is_ball(self):
+    def is_ball(self) -> bool:
         r"""
         Return ``True`` if the subword complex ``self`` is a ball.
 
@@ -1499,7 +1499,7 @@ def is_ball(self):
         """
         return not self.is_sphere()
 
-    def is_pure(self):
+    def is_pure(self) -> bool:
         r"""
         Return ``True`` since all subword complexes are pure.
 
@@ -1541,7 +1541,7 @@ def dimension(self):
     # root and weight
 
     @cached_method
-    def is_root_independent(self):
+    def is_root_independent(self) -> bool:
         r"""
         Return ``True`` if ``self`` is root-independent.
 
@@ -1569,7 +1569,7 @@ def is_root_independent(self):
         return M.rank() == max(M.ncols(), M.nrows())
 
     @cached_method
-    def is_double_root_free(self):
+    def is_double_root_free(self) -> bool:
         r"""
         Return ``True`` if ``self`` is double-root-free.
 
diff --git a/src/sage/dynamics/arithmetic_dynamics/wehlerK3.py b/src/sage/dynamics/arithmetic_dynamics/wehlerK3.py
index 5a093784996..759c5758488 100644
--- a/src/sage/dynamics/arithmetic_dynamics/wehlerK3.py
+++ b/src/sage/dynamics/arithmetic_dynamics/wehlerK3.py
@@ -142,7 +142,7 @@ def __init__(self, polys):
         vars = R.variable_names()
         A = ProductProjectiveSpaces([2, 2],R.base_ring(),vars)
         CR = A.coordinate_ring()
-        #Check for following:
+        # Check for following:
         #    Is the user calling in 2 polynomials from a list or tuple?
         #    Is there one biquadratic and one bilinear polynomial?
         if len(polys) != 2:
@@ -411,7 +411,7 @@ def Hpoly(self, component, i, j):
             sage: X.Hpoly(0, 1, 0)
              2*y0*y1^3 + 2*y0*y1*y2^2 - y1*y2^3
         """
-        #Check Errors in Passed in Values
+        # Check Errors in Passed in Values
         if component not in [0, 1]:
             raise ValueError("component can only be 1 or 0")
 
@@ -709,7 +709,7 @@ def Ramification_poly(self, i):
              4*((self._Lcoeff(i, 2))**2)*(self._Qcoeff(i, 1, 1))*(self._Qcoeff(i, 0, 0))
 
     @cached_method
-    def is_degenerate(self):
+    def is_degenerate(self) -> bool:
         r"""
         Function will return ``True`` if there is a fiber (over the algebraic closure of the
         base ring) of dimension greater than 0 and ``False`` otherwise.
@@ -751,7 +751,7 @@ def is_degenerate(self):
         PP = self.ambient_space()
         K = FractionField(PP[0].base_ring())
         R = PP.coordinate_ring()
-        PS = PP[0] #check for x fibers
+        PS = PP[0]  # check for x fibers
         vars = list(PS.gens())
         R0 = PolynomialRing(K, 3, vars) #for dimension calculation to work,
             #must be done with Polynomial ring over a field
@@ -763,7 +763,7 @@ def is_degenerate(self):
         if I.dimension() != 0:
             return True
 
-        PS = PP[1] #check for y fibers
+        PS = PP[1]  # check for y fibers
         vars = list(PS.gens())
         R0 = PolynomialRing(K,3,vars) #for dimension calculation to work,
         #must be done with Polynomial ring over a field
@@ -840,7 +840,7 @@ def degenerate_fibers(self):
         phi = R.hom(vars + [0, 0, 0], R0)
         I = phi(I)
         xFibers = []
-        #check affine charts
+        # check affine charts
         for n in range(3):
             affvars = list(R0.gens())
             del affvars[n]
@@ -871,7 +871,7 @@ def degenerate_fibers(self):
         phi = PP.coordinate_ring().hom([0, 0, 0] + vars, R0)
         I = phi(I)
         yFibers = []
-        #check affine charts
+        # check affine charts
         for n in range(3):
             affvars = list(R0.gens())
             del affvars[n]
@@ -898,7 +898,7 @@ def degenerate_fibers(self):
     @cached_method
     def degenerate_primes(self, check=True):
         r"""
-        Determine which primes `p` ``self`` has degenerate fibers over `\GF{p}`.
+        Determine primes `p` such that ``self`` has degenerate fibers over `\GF{p}`.
 
         If ``check`` is ``False``, then may return primes that do not have degenerate fibers.
         Raises an error if the surface is degenerate.
@@ -995,7 +995,7 @@ def degenerate_primes(self, check=True):
             if power == 1:
                 bad_primes = bad_primes+GB[i].lt().coefficients()[0].support()
         bad_primes = sorted(set(bad_primes))
-        #check to return only the truly bad primes
+        # check to return only the truly bad primes
         if check:
             for p in bad_primes:
                 X = self.change_ring(GF(p))
@@ -1041,7 +1041,7 @@ def is_smooth(self) -> bool:
         R = self.ambient_space().coordinate_ring()
         I = R.ideal(M.minors(2) + [self.L,self.Q])
         T = PolynomialRing(self.ambient_space().base_ring().fraction_field(), 4, 'h')
-        #check the 9 affine charts for a singular point
+        # check the 9 affine charts for a singular point
         for l in xmrange([3, 3]):
             vars = list(T.gens())
             vars.insert(l[0], 1)
@@ -1577,7 +1577,7 @@ def phi(self, a, **kwds):
             (-1 : 0 : 1 , 0 : 1 : 0)
         """
         A = self.sigmaX(a, **kwds)
-        kwds.update({"check":False})
+        kwds.update({"check": False})
         return self.sigmaY(A, **kwds)
 
     def psi(self, a, **kwds):
@@ -1617,7 +1617,7 @@ def psi(self, a, **kwds):
             (0 : 0 : 1 , 0 : 1 : 0)
         """
         A = self.sigmaY(a, **kwds)
-        kwds.update({"check":False})
+        kwds.update({"check": False})
         return self.sigmaX(A, **kwds)
 
     def lambda_plus(self, P, v, N, m, n, prec=100):
@@ -2411,9 +2411,9 @@ def orbit_psi(self, P, N, **kwds):
             Orb.append(Q)
         return Orb
 
-    def is_isomorphic(self, right):
+    def is_isomorphic(self, right) -> bool:
         r"""
-        Check to see if two K3 surfaces have the same defining ideal.
+        Check whether two K3 surfaces have the same defining ideal.
 
         INPUT:
 
@@ -2449,10 +2449,12 @@ def is_isomorphic(self, right):
         """
         return self.defining_ideal() == right.defining_ideal()
 
-    def is_symmetric_orbit(self, orbit):
+    def is_symmetric_orbit(self, orbit) -> bool:
         r"""
-        Check to see if the orbit is symmetric (i.e. if one of the points on the
-        orbit is fixed by '\sigma_x' or '\sigma_y').
+        Check whether the orbit is symmetric.
+
+        This means that one of the points on the
+        orbit is fixed by '\sigma_x' or '\sigma_y'.
 
         INPUT:
 
diff --git a/src/sage/geometry/hyperplane_arrangement/arrangement.py b/src/sage/geometry/hyperplane_arrangement/arrangement.py
index f807025956d..3f8f47bad37 100644
--- a/src/sage/geometry/hyperplane_arrangement/arrangement.py
+++ b/src/sage/geometry/hyperplane_arrangement/arrangement.py
@@ -1454,7 +1454,7 @@ def has_good_reduction(self, p) -> bool:
         q = a.intersection_poset()
         return p.is_isomorphic(q)
 
-    def is_linear(self):
+    def is_linear(self) -> bool:
         r"""
         Test whether all hyperplanes pass through the origin.
 
@@ -1478,7 +1478,7 @@ def is_linear(self):
         """
         return all(hyperplane.b() == 0 for hyperplane in self)
 
-    def is_essential(self):
+    def is_essential(self) -> bool:
         r"""
         Test whether the hyperplane arrangement is essential.
 
@@ -3016,7 +3016,7 @@ def whitney_number(self, k, kind=1):
                 return self.whitney_data()[1][0, k]
         raise ValueError('argument out of range')
 
-    def is_separating_hyperplane(self, region1, region2, hyperplane):
+    def is_separating_hyperplane(self, region1, region2, hyperplane) -> bool:
         r"""
         Test whether the ``hyperplane`` separates the given regions.
 
@@ -3320,7 +3320,7 @@ def minimal_generated_number(self):
                 return i
         return self.n_hyperplanes()
 
-    def is_formal(self):
+    def is_formal(self) -> bool:
         """
         Return if ``self`` is formal.
 
@@ -3410,7 +3410,7 @@ def derivation_module_free_chain(self):
         return construct_free_chain(self)
 
     @cached_method(key=lambda self, a: None)
-    def is_free(self, algorithm='singular'):
+    def is_free(self, algorithm='singular') -> bool:
         r"""
         Return if ``self`` is free.
 
diff --git a/src/sage/graphs/generic_graph.py b/src/sage/graphs/generic_graph.py
index 3383c2bd44b..9e026298c7b 100644
--- a/src/sage/graphs/generic_graph.py
+++ b/src/sage/graphs/generic_graph.py
@@ -14177,7 +14177,7 @@ def degree_sequence(self):
         """
         return sorted(self.degree_iterator(), reverse=True)
 
-    def is_regular(self, k=None):
+    def is_regular(self, k=None) -> bool:
         """
         Check whether this graph is (`k`-)regular.
 
diff --git a/src/sage/groups/perm_gps/permgroup.py b/src/sage/groups/perm_gps/permgroup.py
index b2adea8b740..934ad843a26 100644
--- a/src/sage/groups/perm_gps/permgroup.py
+++ b/src/sage/groups/perm_gps/permgroup.py
@@ -1336,7 +1336,7 @@ def gen(self, i=None):
                 raise ValueError("you must specify which generator you want")
         return gens[i]
 
-    def ngens(self):
+    def ngens(self) -> int:
         """
         Return the number of generators of ``self``.
 
@@ -4260,11 +4260,12 @@ def is_cyclic(self) -> bool:
         """
         return bool(self._libgap_().IsCyclic())
 
-    def is_elementary_abelian(self):
+    def is_elementary_abelian(self) -> bool:
         """
-        Return ``True`` if this group is elementary abelian. An elementary
-        abelian group is a finite abelian group, where every nontrivial
-        element has order `p`, where `p` is a prime.
+        Return ``True`` if this group is elementary abelian.
+
+        An elementary abelian group is a finite abelian group, where
+        every nontrivial element has order `p`, where `p` is a prime.
 
         EXAMPLES::
 
@@ -4327,7 +4328,7 @@ def isomorphism_to(self, right):
         from .permgroup_morphism import PermutationGroupMorphism_im_gens
         return PermutationGroupMorphism_im_gens(self, right, dsts)
 
-    def is_isomorphic(self, right):
+    def is_isomorphic(self, right) -> bool:
         """
         Return ``True`` if the groups are isomorphic.
 
@@ -4419,7 +4420,7 @@ def is_perfect(self) -> bool:
         """
         return bool(self._libgap_().IsPerfectGroup())
 
-    def is_pgroup(self) -> bools:
+    def is_pgroup(self) -> bool:
         r"""
         Return ``True`` if this group is a `p`-group.
 
@@ -4466,7 +4467,7 @@ def is_simple(self) -> bool:
         """
         return bool(self._libgap_().IsSimpleGroup())
 
-    def is_solvable(self):
+    def is_solvable(self) -> bool:
         """
         Return ``True`` if the group is solvable.
 
@@ -4478,7 +4479,7 @@ def is_solvable(self):
         """
         return bool(self._libgap_().IsSolvableGroup())
 
-    def is_subgroup(self, other):
+    def is_subgroup(self, other) -> bool:
         """
         Return ``True`` if ``self`` is a subgroup of ``other``.
 
@@ -4489,9 +4490,9 @@ def is_subgroup(self, other):
             sage: G.is_subgroup(H)
             True
         """
-        return all((x in other) for x in self.gens())
+        return all(x in other for x in self.gens())
 
-    def is_supersolvable(self):
+    def is_supersolvable(self) -> bool:
         """
         Return ``True`` if the group is supersolvable.
 
@@ -4548,7 +4549,7 @@ def fixed_points(self):
         non_fixed_points = self.non_fixed_points()
         return [i for i in self.domain() if i not in non_fixed_points]
 
-    def is_transitive(self, domain=None):
+    def is_transitive(self, domain=None) -> bool:
         r"""
         Return ``True`` if ``self`` acts transitively on ``domain``.
 
@@ -4604,7 +4605,7 @@ def is_transitive(self, domain=None):
 
         return bool(self._libgap_().IsTransitive(domain))
 
-    def is_primitive(self, domain=None):
+    def is_primitive(self, domain=None) -> bool:
         r"""
         Return ``True`` if ``self`` acts primitively on ``domain``.
 
@@ -4657,7 +4658,7 @@ def is_primitive(self, domain=None):
 
         return bool(self._libgap_().IsPrimitive(domain))
 
-    def is_semi_regular(self, domain=None):
+    def is_semi_regular(self, domain=None) -> bool:
         r"""
         Return ``True`` if ``self`` acts semi-regularly on ``domain``.
 
@@ -4689,7 +4690,7 @@ def is_semi_regular(self, domain=None):
             return False
         return bool(self._libgap_().IsSemiRegular(domain))
 
-    def is_regular(self, domain=None):
+    def is_regular(self, domain=None) -> bool:
         r"""
         Return ``True`` if ``self`` acts regularly on ``domain``.
 
@@ -5270,11 +5271,12 @@ def ambient_group(self):
         """
         return self._ambient_group
 
-    def is_normal(self, other=None):
+    def is_normal(self, other=None) -> bool:
         """
-        Return ``True`` if this group is a normal subgroup of
-        ``other``.  If ``other`` is not specified, then it is assumed
-        to be the ambient group.
+        Return ``True`` if this group is a normal subgroup of ``other``.
+
+        If ``other`` is not specified, then it is assumed to be the
+        ambient group.
 
         EXAMPLES::
 
diff --git a/src/sage/matrix/matrix0.pyx b/src/sage/matrix/matrix0.pyx
index 1db2850130f..42bdca96bfa 100644
--- a/src/sage/matrix/matrix0.pyx
+++ b/src/sage/matrix/matrix0.pyx
@@ -497,7 +497,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         """
         self._is_immutable = True
 
-    def is_immutable(self):
+    def is_immutable(self) -> bool:
         """
         Return ``True`` if this matrix is immutable.
 
@@ -515,7 +515,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         """
         return self._is_immutable
 
-    def is_mutable(self):
+    def is_mutable(self) -> bool:
         """
         Return ``True`` if this matrix is mutable.
 
@@ -4136,7 +4136,7 @@ cdef class Matrix(sage.structure.element.Matrix):
     # Predicates
     ###################################################
 
-    def is_symmetric(self):
+    def is_symmetric(self) -> bool:
         """
         Return ``True`` if this is a symmetric matrix.
 
@@ -4316,7 +4316,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         self.cache(key, True)
         return True
 
-    def is_hermitian(self):
+    def is_hermitian(self) -> bool:
         r"""
         Return ``True`` if the matrix is equal to its conjugate-transpose.
 
@@ -4393,7 +4393,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         """
         return self._is_hermitian(skew=False, tolerance=0)
 
-    def is_skew_hermitian(self):
+    def is_skew_hermitian(self) -> bool:
         r"""
         Return ``True`` if the matrix is equal to the negative of its
         conjugate transpose.
@@ -4452,7 +4452,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         """
         return self._is_hermitian(skew=True, tolerance=0)
 
-    def is_skew_symmetric(self):
+    def is_skew_symmetric(self) -> bool:
         """
         Return ``True`` if ``self`` is a skew-symmetric matrix.
 
@@ -4500,7 +4500,7 @@ cdef class Matrix(sage.structure.element.Matrix):
                     return False
         return True
 
-    def is_alternating(self):
+    def is_alternating(self) -> bool:
         """
         Return ``True`` if ``self`` is an alternating matrix.
 
@@ -4656,7 +4656,7 @@ cdef class Matrix(sage.structure.element.Matrix):
             raise ValueError("The matrix is not a square matrix")
         return self._check_symmetrizability(return_diag=return_diag, skew=True, positive=positive)
 
-    def is_dense(self):
+    def is_dense(self) -> bool:
         """
         Return ``True`` if this is a dense matrix.
 
@@ -4672,7 +4672,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         """
         return self.is_dense_c()
 
-    def is_sparse(self):
+    def is_sparse(self) -> bool:
         """
         Return ``True`` if this is a sparse matrix.
 
@@ -4688,10 +4688,11 @@ cdef class Matrix(sage.structure.element.Matrix):
         """
         return self.is_sparse_c()
 
-    def is_square(self):
+    def is_square(self) -> bool:
         """
-        Return ``True`` precisely if this matrix is square, i.e., has the same
-        number of rows and columns.
+        Return ``True`` precisely if this matrix is square.
+
+        This means that it has the same number of rows and columns.
 
         EXAMPLES::
 
@@ -4702,7 +4703,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         """
         return self._nrows == self._ncols
 
-    def is_invertible(self):
+    def is_invertible(self) -> bool:
         r"""
         Return ``True`` if this matrix is invertible.
 
@@ -4754,7 +4755,7 @@ cdef class Matrix(sage.structure.element.Matrix):
 
     is_unit = is_invertible
 
-    def is_singular(self):
+    def is_singular(self) -> bool:
         r"""
         Return ``True`` if ``self`` is singular.
 
diff --git a/src/sage/rings/number_field/number_field_rel.py b/src/sage/rings/number_field/number_field_rel.py
index 18be930317a..261c8a9ef51 100644
--- a/src/sage/rings/number_field/number_field_rel.py
+++ b/src/sage/rings/number_field/number_field_rel.py
@@ -467,7 +467,7 @@ def subfields(self, degree=0, name=None):
         ans = Sequence(ans, immutable=True, cr=bool(ans))
         return ans
 
-    def is_absolute(self):
+    def is_absolute(self) -> bool:
         r"""
         Return ``False``, since this is not an absolute field.
 
@@ -1171,14 +1171,15 @@ def _pari_base_nf(self):
             sage: k._pari_base_nf()
             [y^2 + 2, [0, 1], -8, 1, ..., [1, 0, 0, -2; 0, 1, 1, 0]]
         """
-        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
+        abs_base, _, _ = self.absolute_base_field()
         return abs_base.pari_nf()
 
-    def is_galois(self):
+    def is_galois(self) -> bool:
         r"""
         For a relative number field, :meth:`is_galois` is deliberately not
         implemented, since it is not clear whether this would mean "Galois over
         `\QQ`" or "Galois over the given base field".
+
         Use either :meth:`is_galois_absolute` or :meth:`is_galois_relative`, respectively.
 
         EXAMPLES::
@@ -1193,7 +1194,7 @@ def is_galois(self):
         """
         raise NotImplementedError("For a relative number field L you must use either L.is_galois_relative() or L.is_galois_absolute() as appropriate")
 
-    def is_galois_relative(self):
+    def is_galois_relative(self) -> bool:
         r"""
         Return ``True`` if for this relative extension `L/K`, `L` is a
         Galois extension of `K`.
@@ -1223,7 +1224,7 @@ def is_galois_relative(self):
             rel_poly = self.relative_polynomial()
             return d == len(rel_poly.base_extend(self).factor())
 
-    def is_galois_absolute(self):
+    def is_galois_absolute(self) -> bool:
         r"""
         Return ``True`` if for this relative extension `L/K`, `L` is a Galois extension of `\QQ`.
 
@@ -1238,7 +1239,7 @@ def is_galois_absolute(self):
         f = self.absolute_polynomial()
         return f.galois_group(pari_group=True).order() == self.absolute_degree()
 
-    def is_isomorphic_relative(self, other, base_isom=None):
+    def is_isomorphic_relative(self, other, base_isom=None) -> bool:
         r"""
         For this relative extension `L/K` and another relative extension `M/K`, return ``True``
         if there is a `K`-linear isomorphism from `L` to `M`. More generally, ``other`` can be a
@@ -1325,10 +1326,12 @@ def is_isomorphic_relative(self, other, base_isom=None):
             raise ValueError("base_isom is not a homomorphism from self's base_field to other's base_field")
         raise ValueError("other must be a relative number field.")
 
-    def is_CM_extension(self):
+    def is_CM_extension(self) -> bool:
         """
-        Return ``True`` is this is a CM extension, i.e. a totally imaginary
-        quadratic extension of a totally real field.
+        Return ``True`` is this is a CM extension.
+
+        This means a totally imaginary quadratic extension of a
+        totally real field.
 
         EXAMPLES::
 
@@ -2481,11 +2484,11 @@ def order(self, *gens, **kwds):
         gens = [self(x) for x in gens]
         return relative_order_from_ring_generators(gens, **kwds)
 
-    def is_free(self, proof=None):
+    def is_free(self, proof=None) -> bool:
         r"""
         Determine whether or not `L/K` is free.
 
-        (i.e. if `\mathcal{O}_L` is a free `\mathcal{O}_K`-module).
+        This means that `\mathcal{O}_L` is a free `\mathcal{O}_K`-module.
 
         INPUT:
 
diff --git a/src/sage/schemes/curves/curve.py b/src/sage/schemes/curves/curve.py
index be072ba3873..81434007b40 100644
--- a/src/sage/schemes/curves/curve.py
+++ b/src/sage/schemes/curves/curve.py
@@ -351,7 +351,7 @@ def singular_points(self, F=None):
         X = self.singular_subscheme()
         return [self.point(p, check=False) for p in X.rational_points(F=F)]
 
-    def is_singular(self, P=None):
+    def is_singular(self, P=None) -> bool:
         r"""
         Return whether ``P`` is a singular point of this curve, or if no point
         is passed, whether this curve is singular or not.
diff --git a/src/sage/schemes/curves/point.py b/src/sage/schemes/curves/point.py
index 782fc4abe14..35759171c91 100644
--- a/src/sage/schemes/curves/point.py
+++ b/src/sage/schemes/curves/point.py
@@ -42,7 +42,7 @@ class ProjectiveCurvePoint_field(SchemeMorphism_point_projective_field):
     """
     Point of a projective curve over a field.
     """
-    def is_singular(self):
+    def is_singular(self) -> bool:
         r"""
         Return whether this point is a singular point of the projective curve it is on.
 
@@ -98,7 +98,7 @@ def tangents(self):
         """
         return self.codomain().tangents(self)
 
-    def is_ordinary_singularity(self):
+    def is_ordinary_singularity(self) -> bool:
         r"""
         Return whether this point is an ordinary singularity of the projective
         plane curve it is on.
@@ -127,7 +127,7 @@ def is_ordinary_singularity(self):
         """
         return self.codomain().is_ordinary_singularity(self)
 
-    def is_transverse(self, D):
+    def is_transverse(self, D) -> bool:
         r"""
         Return whether the intersection of the curve ``D`` at this point with
         the curve this point is on is transverse or not.
@@ -250,7 +250,7 @@ class IntegralProjectivePlaneCurvePoint_finite_field(ProjectivePlaneCurvePoint_f
 
 class AffineCurvePoint_field(SchemeMorphism_point_affine_field):
 
-    def is_singular(self):
+    def is_singular(self) -> bool:
         r"""
         Return whether this point is a singular point of the affine curve it is on.
 
@@ -311,7 +311,7 @@ def tangents(self):
         """
         return self.codomain().tangents(self)
 
-    def is_ordinary_singularity(self):
+    def is_ordinary_singularity(self) -> bool:
         r"""
         Return whether this point is an ordinary singularity of the affine
         plane curve it is on.
@@ -336,7 +336,7 @@ def is_ordinary_singularity(self):
         """
         return self.codomain().is_ordinary_singularity(self)
 
-    def is_transverse(self, D):
+    def is_transverse(self, D) -> bool:
         r"""
         Return whether the intersection of the curve ``D`` at this point with
         the curve this point is on is transverse or not.
diff --git a/src/sage/schemes/curves/projective_curve.py b/src/sage/schemes/curves/projective_curve.py
index 56871de729a..4c3e644145b 100644
--- a/src/sage/schemes/curves/projective_curve.py
+++ b/src/sage/schemes/curves/projective_curve.py
@@ -817,7 +817,7 @@ def plot(self, *args, **kwds):
         C = Curve(self.affine_patch(patch))
         return C.plot(*args, **kwds)
 
-    def is_singular(self, P=None):
+    def is_singular(self, P=None) -> bool:
         r"""
         Return whether this curve is singular or not, or if a point ``P`` is
         provided, whether ``P`` is a singular point of this curve.
@@ -992,7 +992,7 @@ def tangents(self, P, factor=True):
         phi = H(G)
         return [phi(g).homogenize(x) for g in L]
 
-    def is_ordinary_singularity(self, P):
+    def is_ordinary_singularity(self, P) -> bool:
         r"""
         Return whether the singular point ``P`` of this projective plane curve is an ordinary singularity.
 
@@ -1523,7 +1523,7 @@ def extension(self):
                         pts.append(pt)
         return phi
 
-    def is_transverse(self, C, P):
+    def is_transverse(self, C, P) -> bool:
         r"""
         Return whether the intersection of this curve with the curve ``C`` at the point ``P`` is transverse.
 
@@ -1652,7 +1652,7 @@ def arithmetic_genus(self):
         """
         return 1 - self.defining_ideal().hilbert_polynomial()(0)
 
-    def is_complete_intersection(self):
+    def is_complete_intersection(self) -> bool:
         r"""
         Return whether this projective curve is a complete intersection.
 
diff --git a/src/sage/schemes/toric/sheaf/klyachko.py b/src/sage/schemes/toric/sheaf/klyachko.py
index 0513edd70c3..f2d3629a9c9 100644
--- a/src/sage/schemes/toric/sheaf/klyachko.py
+++ b/src/sage/schemes/toric/sheaf/klyachko.py
@@ -739,7 +739,7 @@ def __richcmp__(self, other, op):
 
         return richcmp(self._filt, other._filt, op)
 
-    def is_isomorphic(self, other):
+    def is_isomorphic(self, other) -> bool:
         """
         Test whether two bundles are isomorphic.
 
diff --git a/src/sage/schemes/toric/variety.py b/src/sage/schemes/toric/variety.py
index 289246716cb..8bf6eb4c3d7 100644
--- a/src/sage/schemes/toric/variety.py
+++ b/src/sage/schemes/toric/variety.py
@@ -1232,7 +1232,7 @@ def dimension_singularities(self):
                 return self.dimension() - codim
         return -1
 
-    def is_homogeneous(self, polynomial):
+    def is_homogeneous(self, polynomial) -> bool:
         r"""
         Check if ``polynomial`` is homogeneous.
 
@@ -1307,7 +1307,7 @@ def is_homogeneous(self, polynomial):
                 return False
         return True
 
-    def is_isomorphic(self, another):
+    def is_isomorphic(self, another) -> bool:
         r"""
         Check if ``self`` is isomorphic to ``another``.
 
diff --git a/src/sage/topology/cell_complex.py b/src/sage/topology/cell_complex.py
index e2276fb8839..f69522b7c33 100644
--- a/src/sage/topology/cell_complex.py
+++ b/src/sage/topology/cell_complex.py
@@ -721,7 +721,7 @@ def betti(self, dim=None, subcomplex=None):
         else:
             return dic
 
-    def is_acyclic(self, base_ring=ZZ):
+    def is_acyclic(self, base_ring=ZZ) -> bool:
         """
         Return ``True`` if the reduced homology with coefficients in
         ``base_ring`` of this cell complex is zero.
@@ -1104,7 +1104,7 @@ def graph(self):
         """
         raise NotImplementedError
 
-    def is_connected(self):
+    def is_connected(self) -> bool:
         """
         Return ``True`` if this cell complex is connected.
 
diff --git a/src/sage/topology/cubical_complex.py b/src/sage/topology/cubical_complex.py
index 5df0a809e56..bfdb16d4c60 100644
--- a/src/sage/topology/cubical_complex.py
+++ b/src/sage/topology/cubical_complex.py
@@ -1015,7 +1015,7 @@ def __hash__(self):
         """
         return hash(frozenset(self._facets))
 
-    def is_subcomplex(self, other):
+    def is_subcomplex(self, other) -> bool:
         r"""
         Return ``True`` if ``self`` is a subcomplex of ``other``.
 
diff --git a/src/sage/topology/simplicial_complex.py b/src/sage/topology/simplicial_complex.py
index b6b58d72e98..fb4fc213421 100644
--- a/src/sage/topology/simplicial_complex.py
+++ b/src/sage/topology/simplicial_complex.py
@@ -431,7 +431,7 @@ def set(self):
         """
         return self.__set
 
-    def is_face(self, other):
+    def is_face(self, other) -> bool:
         """
         Return ``True`` iff this simplex is a face of other.
 
@@ -552,7 +552,7 @@ def dimension(self):
         """
         return len(self.__tuple) - 1
 
-    def is_empty(self):
+    def is_empty(self) -> bool:
         """
         Return ``True`` iff this simplex is the empty simplex.
 
@@ -1410,7 +1410,7 @@ def face_iterator(self, increasing=True):
 
     n_faces = GenericCellComplex.n_cells
 
-    def is_pure(self):
+    def is_pure(self) -> bool:
         """
         Return ``True`` iff this simplicial complex is pure.
 
@@ -1718,7 +1718,7 @@ def flip_graph(self):
                 edges[coF].append(F)
         return flipG
 
-    def is_pseudomanifold(self):
+    def is_pseudomanifold(self) -> bool:
         """
         Return ``True`` if ``self`` is a pseudomanifold.
 
@@ -2843,7 +2843,7 @@ def remove_faces(self, faces, check=False):
         for f in faces:
             self.remove_face(f, check=check)
 
-    def is_subcomplex(self, other):
+    def is_subcomplex(self, other) -> bool:
         """
         Return ``True`` if this is a subcomplex of ``other``.
 
@@ -3001,7 +3001,7 @@ def star(self, simplex, is_mutable=True):
                 faces.append(f)
         return SimplicialComplex(faces, is_mutable=is_mutable)
 
-    def is_cohen_macaulay(self, base_ring=QQ, ncpus=0):
+    def is_cohen_macaulay(self, base_ring=QQ, ncpus=0) -> bool:
         r"""
         Return ``True`` if ``self`` is Cohen-Macaulay.
 
@@ -3102,7 +3102,7 @@ def generated_subcomplex(self, sub_vertex_set, is_mutable=True):
         return SimplicialComplex(faces, maximality_check=True,
                                  is_mutable=is_mutable)
 
-    def is_shelling_order(self, shelling_order, certificate=False):
+    def is_shelling_order(self, shelling_order, certificate=False) -> bool:
         r"""
         Return if the order of the facets given by ``shelling_order``
         is a shelling order for ``self``.
@@ -3172,7 +3172,7 @@ def is_shelling_order(self, shelling_order, certificate=False):
         return True
 
     @cached_method
-    def is_shellable(self, certificate=False):
+    def is_shellable(self, certificate=False) -> bool:
         r"""
         Return if ``self`` is shellable.
 
@@ -3800,7 +3800,7 @@ def delta_complex(self, sort_simplices=False):
             n_cells = bdries
         return DeltaComplex(data)
 
-    def is_flag_complex(self):
+    def is_flag_complex(self) -> bool:
         """
         Return ``True`` if and only if ``self`` is a flag complex.
 
@@ -4208,7 +4208,7 @@ def fundamental_group(self, base_point=None, simplify=True):
         else:
             return FG.quotient(rels)
 
-    def is_isomorphic(self, other, certificate=False):
+    def is_isomorphic(self, other, certificate=False) -> bool:
         r"""
         Check whether two simplicial complexes are isomorphic.
 
@@ -4555,7 +4555,7 @@ def set_immutable(self):
         self._is_immutable = True
         self._facets = tuple(self._facets)
 
-    def is_mutable(self):
+    def is_mutable(self) -> bool:
         """
         Return ``True`` if mutable.
 
@@ -4576,7 +4576,7 @@ def is_mutable(self):
         """
         return not self._is_immutable
 
-    def is_immutable(self):
+    def is_immutable(self) -> bool:
         """
         Return ``True`` if immutable.
 
@@ -4591,7 +4591,7 @@ def is_immutable(self):
         """
         return self._is_immutable
 
-    def cone_vertices(self):
+    def cone_vertices(self) -> list:
         r"""
         Return the list of cone vertices of ``self``.
 
@@ -4632,7 +4632,7 @@ def decone(self):
         V = set(self.vertices()).difference(self.cone_vertices())
         return self.generated_subcomplex(V)
 
-    def is_balanced(self, check_purity=False, certificate=False):
+    def is_balanced(self, check_purity=False, certificate=False) -> bool:
         r"""
         Determine whether ``self`` is balanced.
 
@@ -5004,6 +5004,8 @@ def is_golod(self) -> bool:
         Massey operations in the associated Tor-algebra are trivial. This
         is done by checking the bigraded Betti numbers.
 
+        .. SEEALSO:: :meth:`is_minimally_non_golod`
+
         EXAMPLES::
 
             sage: # needs sage.modules
@@ -5014,11 +5016,11 @@ def is_golod(self) -> bool:
             sage: Y.is_golod()
             True
         """
-        H = [a+b for a, b in self.bigraded_betti_numbers()]
+        H = [a + b for a, b in self.bigraded_betti_numbers()]
         if 0 in H:
             H.remove(0)
 
-        return not any(i+j in H for ii, i in enumerate(H) for j in H[ii:])
+        return not any(i + j in H for ii, i in enumerate(H) for j in H[ii:])
 
     def is_minimally_non_golod(self) -> bool:
         r"""