gh-35097: Simplicial set group
    
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### 📚 Implementation of finite covers of simplicial sets

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Compute the cover of a (based) simplicial set associated to a
representation of its fundamental group to a finite group.

Fixes #34886.

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### ⌛ Dependencies
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URL: #35097
Reported by: miguelmarco
Reviewer(s): John H. Palmieri

Author: Release Manager

src/sage/categories/simplicial_sets.py

@@ -317,7 +317,7 @@ def fundamental_group(self, simplify=True):
                     sage: Sigma3 = groups.permutation.Symmetric(3)
                     sage: BSigma3 = Sigma3.nerve()
                     sage: pi = BSigma3.fundamental_group(); pi
-                    Finitely presented group < e0, e1 | e0^2, e1^3, (e0*e1^-1)^2 >
+                    Finitely presented group < e1, e2 | e2^2, e1^3, (e2*e1)^2 >
                     sage: pi.order()
                     6
                     sage: pi.is_abelian()
@@ -331,19 +331,35 @@ def fundamental_group(self, simplify=True):
                 """
                 # Import this here to prevent importing libgap upon startup.
                 from sage.groups.free_group import FreeGroup
-                skel = self.n_skeleton(2)
+                if not self.n_cells(1):
+                    return FreeGroup([]).quotient([])
+                FG = self._universal_cover_dict()[0]
+                if simplify:
+                    return FG.simplified()
+                else:
+                    return FG
 
+            def _universal_cover_dict(self):
+                r"""
+                Return the fundamental group and dictionary sending each edge to
+                the corresponding group element
+
+                TESTS::
+
+                    sage: RP2 = simplicial_sets.RealProjectiveSpace(2)
+                    sage: RP2._universal_cover_dict()
+                    (Finitely presented group < e | e^2 >, {f: e})
+                    sage: RP2.nondegenerate_simplices()
+                    [1, f, f * f]
+                """
+                from sage.groups.free_group import FreeGroup
+                skel = self.n_skeleton(2)
                 graph = skel.graph()
                 if not skel.is_connected():
                     graph = graph.subgraph(skel.base_point())
-
-                edges = [e[2] for e in graph.edges(sort=True)]
+                edges = [e[2] for e in graph.edges(sort=False)]
                 spanning_tree = [e[2] for e in graph.min_spanning_tree()]
                 gens = [e for e in edges if e not in spanning_tree]
-
-                if not gens:
-                    return FreeGroup([]).quotient([])
-
                 gens_dict = dict(zip(gens, range(len(gens))))
                 FG = FreeGroup(len(gens), 'e')
                 rels = []
@@ -361,10 +377,192 @@ def fundamental_group(self, simplify=True):
                             # sigma is not in the correct connected component.
                             z[i] = FG.one()
                     rels.append(z[0]*z[1].inverse()*z[2])
-                if simplify:
-                    return FG.quotient(rels).simplified()
-                else:
-                    return FG.quotient(rels)
+                G = FG.quotient(rels)
+                char = {g : G.gen(i) for i,g in enumerate(gens)}
+                for e in edges:
+                    if e not in gens:
+                        char[e] = G.one()
+                return (G, char)
+
+
+            def universal_cover_map(self):
+                r"""
+                Return the universal covering map of the simplicial set.
+
+                It requires the fundamental group to be finite.
+
+                EXAMPLES::
+
+                    sage: RP2 = simplicial_sets.RealProjectiveSpace(2)
+                    sage: phi = RP2.universal_cover_map()
+                    sage: phi
+                    Simplicial set morphism:
+                      From: Simplicial set with 6 non-degenerate simplices
+                      To:   RP^2
+                      Defn: [(1, 1), (1, e), (f, 1), (f, e), (f * f, 1), (f * f, e)] --> [1, 1, f, f, f * f, f * f]
+                    sage: phi.domain().face_data()
+                        {(1, 1): None,
+                         (1, e): None,
+                         (f, 1): ((1, e), (1, 1)),
+                         (f, e): ((1, 1), (1, e)),
+                         (f * f, 1): ((f, e), s_0 (1, 1), (f, 1)),
+                         (f * f, e): ((f, 1), s_0 (1, e), (f, e))}
+
+                """
+                edges = self.n_cells(1)
+                if not edges:
+                    return self.identity()
+                G, char = self._universal_cover_dict()
+                return self.covering_map(char)
+
+            def covering_map(self, character):
+                r"""
+                Return the covering map associated to a character.
+
+                The character is represented by a dictionary that assigns an
+                element of a finite group to each nondegenerate 1-dimensional
+                cell. It should correspond to an epimorphism from the fundamental
+                group.
+
+                INPUT:
+
+                - ``character`` -- a dictionary
+
+
+                EXAMPLES::
+
+                    sage: S1 = simplicial_sets.Sphere(1)
+                    sage: W = S1.wedge(S1)
+                    sage: G = CyclicPermutationGroup(3)
+                    sage: a, b = W.n_cells(1)
+                    sage: C = W.covering_map({a : G.gen(0), b : G.one()})
+                    sage: C
+                    Simplicial set morphism:
+                      From: Simplicial set with 9 non-degenerate simplices
+                      To:   Wedge: (S^1 v S^1)
+                      Defn: [(*, ()), (*, (1,2,3)), (*, (1,3,2)), (sigma_1, ()), (sigma_1, ()), (sigma_1, (1,2,3)), (sigma_1, (1,2,3)), (sigma_1, (1,3,2)), (sigma_1, (1,3,2))] --> [*, *, *, sigma_1, sigma_1, sigma_1, sigma_1, sigma_1, sigma_1]
+                    sage: C.domain()
+                    Simplicial set with 9 non-degenerate simplices
+                    sage: C.domain().face_data()
+                    {(*, ()): None,
+                     (*, (1,2,3)): None,
+                     (*, (1,3,2)): None,
+                     (sigma_1, ()): ((*, (1,2,3)), (*, ())),
+                     (sigma_1, ()): ((*, ()), (*, ())),
+                     (sigma_1, (1,2,3)): ((*, (1,3,2)), (*, (1,2,3))),
+                     (sigma_1, (1,2,3)): ((*, (1,2,3)), (*, (1,2,3))),
+                     (sigma_1, (1,3,2)): ((*, ()), (*, (1,3,2))),
+                     (sigma_1, (1,3,2)): ((*, (1,3,2)), (*, (1,3,2)))}
+                """
+                from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet
+                from sage.topology.simplicial_set_morphism import SimplicialSetMorphism
+                char = {a : b for (a,b) in character.items()}
+                G = list(char.values())[0].parent()
+                if not G.is_finite():
+                    raise NotImplementedError("can only compute universal covers of spaces with finite fundamental group")
+                cells_dict = {}
+                faces_dict = {}
+
+                for s in self.n_cells(0):
+                    for g in G:
+                        cell =  AbstractSimplex(0,name="({}, {})".format(s, g))
+                        cells_dict[(s,g)] = cell
+                        char[s] = G.one()
+
+                for d in range(1, self.dimension() + 1):
+                    for s in self.n_cells(d):
+                        if s not in char.keys():
+                            if d==1 and  s.is_degenerate():
+                                char[s] = G.one()
+                            elif s.is_degenerate():
+                                if 0 in s.degeneracies():
+                                    char[s] = G.one()
+                                else:
+                                    char[s] = char[s.nondegenerate()]
+                            else:
+                                char[s] = char[self.face(s, d)]
+                        if s.is_nondegenerate():
+                            for g in G:
+                                cell =  AbstractSimplex(d,name="({}, {})".format(s, g))
+                                cells_dict[(s,g)] = cell
+                                fd = []
+                                faces = self.faces(s)
+                                f0 = faces[0]
+                                for h in G:
+                                    if h == g*char[s]:
+                                        lifted = h
+                                        break
+                                grelems = [cells_dict[(f0.nondegenerate(), lifted)].apply_degeneracies(*f0.degeneracies())]
+                                for f in faces[1:]:
+                                    grelems.append(cells_dict[(f.nondegenerate(), g)].apply_degeneracies(*f.degeneracies()))
+                                faces_dict[cell] = grelems
+                cover = SimplicialSet(faces_dict, base_point=cells_dict[(self.base_point(), G.one())])
+                cover_map_data = {c : s[0] for (s,c) in cells_dict.items()}
+                return SimplicialSetMorphism(data = cover_map_data, domain = cover, codomain = self)
+
+            def cover(self, character):
+                r"""
+                Return the cover of the simplicial set associated to a character
+                of the fundamental group.
+
+                The character is represented by a dictionary, that assigns an
+                element of a finite group to each nondegenerate 1-dimensional
+                cell. It should correspond to an epimorphism from the fundamental
+                group.
+
+                INPUT:
+
+                - ``character`` -- a dictionary
+
+                EXAMPLES::
+
+                    sage: S1 = simplicial_sets.Sphere(1)
+                    sage: W = S1.wedge(S1)
+                    sage: G = CyclicPermutationGroup(3)
+                    sage: (a, b) = W.n_cells(1)
+                    sage: C = W.cover({a : G.gen(0), b : G.gen(0)^2})
+                    sage: C.face_data()
+                    {(*, ()): None,
+                     (*, (1,2,3)): None,
+                     (*, (1,3,2)): None,
+                     (sigma_1, ()): ((*, (1,2,3)), (*, ())),
+                     (sigma_1, ()): ((*, (1,3,2)), (*, ())),
+                     (sigma_1, (1,2,3)): ((*, (1,3,2)), (*, (1,2,3))),
+                     (sigma_1, (1,2,3)): ((*, ()), (*, (1,2,3))),
+                     (sigma_1, (1,3,2)): ((*, ()), (*, (1,3,2))),
+                     (sigma_1, (1,3,2)): ((*, (1,2,3)), (*, (1,3,2)))}
+                    sage: C.homology(1)
+                    Z x Z x Z x Z
+                    sage: C.fundamental_group()
+                    Finitely presented group < e0, e1, e2, e3 |  >
+                """
+                return self.covering_map(character).domain()
+
+            def universal_cover(self):
+                r"""
+                Return the universal cover of the simplicial set.
+                The fundamental group must be finite in order to ensure that the
+                universal cover is a simplicial set of finite type.
+
+                EXAMPLES::
+
+                    sage: RP3 = simplicial_sets.RealProjectiveSpace(3)
+                    sage: C = RP3.universal_cover()
+                    sage: C
+                    Simplicial set with 8 non-degenerate simplices
+                    sage: C.face_data()
+                    {(1, 1): None,
+                     (1, e): None,
+                     (f, 1): ((1, e), (1, 1)),
+                     (f, e): ((1, 1), (1, e)),
+                     (f * f, 1): ((f, e), s_0 (1, 1), (f, 1)),
+                     (f * f, e): ((f, 1), s_0 (1, e), (f, e)),
+                     (f * f * f, 1): ((f * f, e), s_0 (f, 1), s_1 (f, 1), (f * f, 1)),
+                     (f * f * f, e): ((f * f, 1), s_0 (f, e), s_1 (f, e), (f * f, e))}
+                    sage: C.fundamental_group()
+                    Finitely presented group <  |  >
+                """
+                return self.universal_cover_map().domain()
 
             def is_simply_connected(self):
                 """

src/sage/topology/simplicial_set.py

@@ -173,7 +173,7 @@
     sage: Sigma3 = groups.permutation.Symmetric(3)
     sage: BSigma3 = Sigma3.nerve()
     sage: pi = BSigma3.fundamental_group(); pi
-    Finitely presented group < e0, e1 | e0^2, e1^3, (e0*e1^-1)^2 >
+    Finitely presented group < e1, e2 | e2^2, e1^3, (e2*e1)^2 >
     sage: pi.order()
     6
     sage: pi.is_abelian()
@@ -1686,26 +1686,12 @@ def graph(self):
             sage: Sigma3.nerve().is_connected()
             True
         """
-        skel = self.n_skeleton(1)
-        edges = skel.n_cells(1)
-        vertices = skel.n_cells(0)
-        used_vertices = set()  # vertices which are in an edge
-        d = {}
-        for e in edges:
-            v = skel.face(e, 0)
-            w = skel.face(e, 1)
-            if v in d:
-                if w in d[v]:
-                    d[v][w] = d[v][w] + [e]
-                else:
-                    d[v][w] = [e]
-            else:
-                d[v] = {w: [e]}
-            used_vertices.update([v, w])
-        for v in vertices:
-            if v not in used_vertices:
-                d[v] = {}
-        return Graph(d, format='dict_of_dicts')
+        G = Graph(loops=True, multiedges=True)
+        for e in self.n_cells(1):
+            G.add_edge(self.face(e,0), self.face(e,1), e)
+        for v in self.n_cells(0):
+            G.add_vertex(v)
+        return G
 
     def is_connected(self):
         """

src/sage/topology/simplicial_set_catalog.py

@@ -26,6 +26,8 @@
 - the Hopf map: this is a pre-built morphism, from which one can
   extract its domain, codomain, mapping cone, etc.
 
+- the complex of a group presentation.
+
 All of these examples are accessible by typing
 ``simplicial_sets.NAME``, where ``NAME`` is the name of the
 example. Type ``simplicial_sets.[TAB]`` for a complete list.
@@ -48,4 +50,4 @@
                                       KleinBottle, Torus,
                                       Simplex, Horn, Point,
                                       ComplexProjectiveSpace,
-                                      HopfMap)
+                                      HopfMap, PresentationComplex)

src/sage/topology/simplicial_set_examples.py

@@ -9,6 +9,8 @@
 AUTHORS:
 
 - John H. Palmieri (2016-07)
+
+- Miguel Marco (2022-12)
 """
 # ****************************************************************************
 #  Copyright (C) 2016 John H. Palmieri <palmieri at math.washington.edu>
@@ -48,6 +50,8 @@
 from sage.misc.lazy_import import lazy_import
 lazy_import('sage.categories.simplicial_sets', 'SimplicialSets')
 
+
+
 ########################################################################
 # The nerve of a finite monoid, used in sage.categories.finite_monoid.
 
@@ -790,3 +794,65 @@ def HopfMap():
     return S3.Hom(S2)({alpha_1:s0_sigma, alpha_2:s1_sigma,
                        alpha_3:s2_sigma, alpha_4:s0_sigma,
                        alpha_5:s2_sigma, alpha_6:s1_sigma})
+
+
+def PresentationComplex(G):
+    r"""
+    Return a simplicial set constructed from a group presentation.
+    The result is a subdivision of the presentation complex.
+
+    The presentation complex has a single vertex and it has one edge for
+    each generator. Then triangles (and eventually new edges
+    to glue them) are added to realize the relations.
+
+    INPUT:
+
+    - "G" -- a finitely presented group
+
+    EXAMPLES::
+
+        sage: G = SymmetricGroup(2).as_finitely_presented_group()
+        sage: G
+        Finitely presented group < a | a^2 >
+        sage: S = simplicial_sets.PresentationComplex(G)
+        sage: S
+        Simplicial set with 5 non-degenerate simplices
+        sage: S.face_data()
+        {Delta^0: None,
+         a: (Delta^0, Delta^0),
+         a^-1: (Delta^0, Delta^0),
+         Ta: (a, s_0 Delta^0, a^-1),
+         a^2: (a, s_0 Delta^0, a)}
+        sage: S.fundamental_group()
+        Finitely presented group < e0 | e0^2 >
+    """
+    O = AbstractSimplex(0)
+    SO = O.apply_degeneracies(0)
+    edges = {g: AbstractSimplex(1, name=str(g)) for g in G.gens()}
+    inverseedges = {g.inverse(): AbstractSimplex(1, name=str(g.inverse())) for g in G.gens()}
+    all_edges = {}
+    all_edges.update(edges)
+    all_edges.update(inverseedges)
+    triangles = {g: AbstractSimplex(2, name='T' + str(g)) for g in G.gens()}
+    face_maps = {g: [O, O] for g in all_edges.values()}
+    face_maps.update({triangles[t]: [all_edges[t], SO, all_edges[t.inverse()]] for t in triangles})
+    for r in G.relations():
+        if len(r.Tietze()) == 1:
+            pass
+        elif len(r.Tietze()) == 2:
+            a = all_edges[G([r.Tietze()[0]])]
+            b = all_edges[G([r.Tietze()[1]])]
+            T = AbstractSimplex(2, name=str(r))
+            face_maps[T] = [a, SO, b]
+        else:
+            words = [all_edges[G([a])] for a in r.Tietze()]
+            words[-1] = all_edges[G([-r.Tietze()[-1]])]
+            while len(words) > 3:
+                auxedge = AbstractSimplex(1)
+                face_maps[auxedge] = [O, O]
+                auxtring = AbstractSimplex(2)
+                face_maps[auxtring] = [words[1], auxedge, words[0]]
+                words = [auxedge] + words[2:]
+            auxtring = AbstractSimplex(2)
+            face_maps[auxtring] = [words[1], words[2], words[0]]
+    return SimplicialSet_finite(face_maps, base_point=O)