gh-36849: Problem with orientations of simplices in simplicial complex maps.
    
Problem with orientation of simplices in simplicial complex maps

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In rare cases, Sage incorrectly computes the chain complex map
associated to a map of simplicial complexes. The problem is that the
orientation of each simplex in the image is not correctly compared to
its actual orientation in the codomain: an edge `(f(v), f(w))` is not
compared correctly to see whether the actual simplex in the codomain is
stored as `(f(v), f(w))` or `(f(w), f(v))`.

I think that the default sorting of vertices (and hence simplices) means
that this usually doesn't come up, but it can arise when the vertices do
not have an obvious ordering. See
https://ask.sagemath.org/question/74754/chain-morphism-between-
subdivisions/ for one such case.

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URL: #36849
Reported by: John H. Palmieri
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/topology/simplicial_complex_morphism.py

@@ -244,20 +244,43 @@ def __call__(self, x, orientation=False):
             (0, 1)
             sage: g(Simplex([0,1]), orientation=True)                                   # needs sage.modules
             ((0, 1), -1)
+
+        TESTS:
+
+        Test that the problem in :issue:`36849` has been fixed::
+
+            sage: S = SimplicialComplex([[1,2]],is_mutable=False).barycentric_subdivision()
+            sage: T = SimplicialComplex([[1,2],[2,3],[1,3]],is_mutable=False).barycentric_subdivision()
+            sage: f = {x[0]:x[0] for x in S.cells()[0]}
+            sage: H = Hom(S,T)
+            sage: z = H(f)
+            sage: z.associated_chain_complex_morphism()
+            Chain complex morphism:
+              From: Chain complex with at most 2 nonzero terms over Integer Ring
+              To:   Chain complex with at most 2 nonzero terms over Integer Ring
         """
         dim = self.domain().dimension()
         if not isinstance(x, Simplex) or x.dimension() > dim or x not in self.domain().faces()[x.dimension()]:
             raise ValueError("x must be a simplex of the source of f")
         tup = x.tuple()
-        fx = []
-        for j in tup:
-            fx.append(self._vertex_dictionary[j])
+        fx = [self._vertex_dictionary[j] for j in tup]
         if orientation:
             from sage.algebras.steenrod.steenrod_algebra_misc import convert_perm
             from sage.combinat.permutation import Permutation
 
             if len(set(fx)) == len(tup):
-                oriented = Permutation(convert_perm(fx)).signature()
+                # We need to compare the image simplex, as given in
+                # the order specified by self, with its orientation in
+                # the codomain.
+                image = Simplex(set(fx))
+                Y_faces = self.codomain()._n_cells_sorted(image.dimension())
+                idx = Y_faces.index(image)
+                actual_image = Y_faces[idx]
+                # The signature of the permutation specified by self:
+                sign_image = Permutation(convert_perm(fx)).signature()
+                # The signature of the permutation of the simplex in the domain:
+                sign_simplex = Permutation(convert_perm(actual_image)).signature()
+                oriented = sign_image * sign_simplex
             else:
                 oriented = 1
             return (Simplex(set(fx)), oriented)