Ticket #13725: trac_13725-sum-complex.patch

sage/homology/examples.py

@@ -40 +40 @@
    simplicial_complexes.RealProjectiveSpace
    simplicial_complexes.Simplex
    simplicial_complexes.Sphere
+   simplicial_complexes.SumComplex
    simplicial_complexes.SurfaceOfGenus
    simplicial_complexes.Torus
 
@@ -170 +171 @@
     - :meth:`RealProjectiveSpace`
     - :meth:`Simplex`
     - :meth:`Sphere`
+    - :meth:`SumComplex`
     - :meth:`SurfaceOfGenus`
     - :meth:`Torus`
 
@@ -1063 +1065 @@
             facets.extend([f for f in maybe if random.random() <= p])
             return SimplicialComplex(facets, is_mutable=False)
 
+    def SumComplex(self, n, A):
+        r"""
+        The sum complexes of Linial, Meshulam, and Rosenthal.
+
+        INPUT:
+
+        - ``n``, a positive integer
+        - ``A``, a subset of `\ZZ/(n)`
+
+        If `k+1` is the cardinality of `A`, then this returns a
+        `k`-dimensional simplicial complex `X_A` with vertices
+        `\ZZ/(n)`, and facets given by all `k+1`-tuples `(x_0, x_1,
+        ..., x_k)` such that the sum `\sum x_i` is in `A`. See the
+        paper by Linial, Meshulam, and Rosenthal [LMR2010]_, in which
+        they prove various results about these complexes; for example,
+        if `n` is prime, then `X_A` is rationally acyclic, and if in
+        addition `A` forms an arithmetic progression in `\ZZ/(n)`,
+        then `X_A` is `\ZZ`-acyclic. Throughout their paper, they
+        assume that `n` and `k` are relatively prime, but the
+        construction makes sense in general.
+
+        In addition to the results from the cited paper, these
+        complexes can have large torsion, given the number of
+        vertices; for example, if `n=10`, and `A=\{0,1,2,3,6\}`, then
+        `H_3(X_A)` is cyclic of order 2728, and there is a
+        4-dimensional complex on 13 vertices with `H_3` having a
+        cyclic summand of order
+
+        .. math::
+    
+            706565607945 = 3 \times 5 \times 53 \times 79 \times 131
+            \times 157 \times 547.
+
+        See the examples.
+
+        REFERENCES:
+
+        .. [LMR2010] N. Linial, R. Meshulam and M. Rosenthal, "Sum
+           complexes -- a new family of hypertrees", Discrete &
+           Computational Geometry, 2010, Volume 44, Number 3, Pages
+           622-636
+
+        EXAMPLES::
+
+            sage: simplicial_complexes.SumComplex(10, [0,1,2,3,6]).homology()
+            {0: 0, 1: 0, 2: 0, 3: C2728, 4: 0}
+            sage: factor(2728)
+            2^3 * 11 * 31
+
+            sage: simplicial_complexes.SumComplex(11, [0, 1, 3]).homology(1)
+            C23
+            sage: simplicial_complexes.SumComplex(11, [0,1,2,3,4,7]).homology() # long time
+            {0: 0, 1: 0, 2: 0, 3: 0, 4: C645679, 5: 0}
+            sage: factor(645679)
+            23 * 67 * 419
+
+            sage: simplicial_complexes.SumComplex(13, [0, 1, 3]).homology(1)
+            C159
+            sage: factor(159)
+            3 * 53
+            sage: simplicial_complexes.SumComplex(13, [0,1,2,5]).homology() # long time
+            {0: 0, 1: 0, 2: C146989209, 3: 0}
+            sage: factor(1648910295)
+            3^2 * 5 * 53 * 521 * 1327
+            sage: simplicial_complexes.SumComplex(13, [0,1,2,3,5]).homology() # long time
+            {0: 0, 1: 0, 2: 0, 3: C3 x C237 x C706565607945, 4: 0}
+            sage: factor(706565607945)
+            3 * 5 * 53 * 79 * 131 * 157 * 547
+
+            sage: simplicial_complexes.SumComplex(17, [0, 1, 4]).homology(1)
+            C140183
+            sage: factor(140183)
+            103 * 1361
+            sage: simplicial_complexes.SumComplex(19, [0, 1, 4]).homology(1)
+            C5670599
+            sage: factor(5670599)
+            11 * 191 * 2699
+            sage: simplicial_complexes.SumComplex(31, [0, 1, 4]).homology(1) # long time
+            C5 x C5 x C5 x C5 x C26951480558170926865
+            sage: factor(26951480558170926865)
+            5 * 311 * 683 * 1117 * 11657 * 1948909
+        """
+        from sage.rings.all import Integers
+        from sage.sets.set import Set
+        from sage.homology.all import SimplicialComplex
+        Zn = Integers(n)
+        A = frozenset([Zn(x) for x in A])
+        k = len(A) - 1
+        facets = []
+        for f in Set(Zn).subsets(k+1):
+            if sum(f) in A:
+                facets.append(tuple(f))
+        return SimplicialComplex(facets)
+
 simplicial_complexes = SimplicialComplexExamples()