.. _hom-comp: .. _homology-computation: Homology Computation ==================== The code computes exclusively reduced homology of the given simplicial complexes. If you want the non-reduced homology, just add a single free rank to dimension 0 and let it be generated by any single point in the complex. .. magma:function:: Homology(X) :input_types: SmpCpx :output_types: SeqEnum, SeqEnum :label: Homology_SmpCpx .. magma:function:: Homology($\sim$X) :input_types: SmpCpx :label: Homology_SmpCpx_ref .. magma:function:: Homology(X,A) :input_types: SmpCpx, Rng :output_types: SeqEnum, SeqEnum :label: Homology_SmpCpx_Rng .. magma:function:: Homology($\sim$X, A) :input_types: SmpCpx, Rng :label: Homology_SmpCpx_Rng_ref Calculates the reduced homology of a simplicial complex ``X`` with coefficients in the ring :math:`A`. The procedural form of this command caches the results of the calculation in the simplicial complex object. If no ring is given, then the function defaults to integer coefficients. .. magma:example:: Example: homology :label: homology The resulting modules are stored in falling dimension, always including the dimension :math:`-1` vanishing homology module at the very end. .. code-block:: magma > circ := Boundary(SimplicialComplex([{1,2,3}])); > Homology(circ,Integers()); [ Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 0 ], Full Quotient RSpace of degree 0 over Integer Ring Column moduli: [ ], Full Quotient RSpace of degree 0 over Integer Ring Column moduli: [ ] ] [ Mapping from: RSpace of degree 3, dimension 1 over Integer Ring to Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 0 ], Mapping from: RSpace of degree 3, dimension 2 over Integer Ring to Full Quotient RSpace of degree 0 over Integer Ring Column moduli: [ ], Mapping from: Full RSpace of degree 1 over Integer Ring to Full Quotient RSpace of degree 0 over Integer Ring Column moduli: [ ] ] > lens3 := LensSpace(3); > Homology(~lens3,Integers()); > Homology(lens3,Integers())[3]; Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 3 ] .. magma:function:: HomologyGroup(X, q) :input_types: SmpCpx, RngIntElt :output_types: ModRng :label: HomologyGroup_SmpCpx_RngIntElt .. magma:function:: HomologyGroup(X, q, A) :input_types: SmpCpx, RngIntElt, Rng :output_types: ModRng :label: HomologyGroup_SmpCpx_RngIntElt_Rng Calculates and returns the :math:`q`\ th homology group of :math:`X` with coefficients in :math:`A`. If no ring is given, the function defaults to integer coefficients. If the homology is cached in :math:`X`, the cached results are returned. This function will not compute the entire homology in order to return one homology group. .. magma:function:: BettiNumber(X,q) :input_types: SmpCpx, RngIntElt :output_types: RngIntElt :label: BettiNumber_SmpCpx_RngIntElt .. magma:function:: BettiNumber(X,q,A) :input_types: SmpCpx, RngIntElt, Rng :output_types: RngIntElt :label: BettiNumber_SmpCpx_RngIntElt_Rng Returns the :math:`q`\ th Betti number, computed as the free rank of the :math:`q`-dimensional homology group, with coefficients in :math:`A`. If no ring is given, then the function will default to integer coefficients. Note that the Betti number computations compensate for the homology computations being reduced. Thus, ``BettiNumber(X,0) eq Rank(HomologyGroup(X,0)) + 1``. .. magma:function:: TorsionCoefficients(X, q) :input_types: SmpCpx, RngIntElt :output_types: SeqEnum[RngElt] :label: TorsionCoefficients_SmpCpx_RngIntElt .. magma:function:: TorsionCoefficients(X, q, A) :input_types: SmpCpx, RngIntElt, Rng :output_types: SeqEnum[RngElt] :label: TorsionCoefficients_SmpCpx_RngIntElt_Rng Returns the torsion coefficients of the :math:`q`\ th homology group of :math:`X` with coefficients in :math:`A`. If no ring is given, then the function will default to integer coefficients. .. magma:function:: EulerCharacteristic(X) :input_types: SmpCpx :output_types: RngIntElt :label: EulerCharacteristic_SmpCpx Computes the Euler characteristic of the complex. If homology is cached, this is used for computation, and else the characteristic is computed using the ranks of the chain groups. .. magma:function:: BoundaryMatrix(X, q, A) :input_types: SmpCpx, RngIntElt, Rng :output_types: Mtrx :label: BoundaryMatrix_SmpCpx_RngIntElt_Rng Returns the :math:`q`\ th boundary matrix of the corresponding chain complex to the simplicial complex :math:`X` with coefficients in :math:`A`. .. magma:function:: ChainComplex(X, A) :input_types: SmpCpx, Rng :output_types: ModCpx :label: ChainComplex_SmpCpx_Rng Constructs a reduced chain complex of free A-modules corresponding to the abstract simplicial complex :math:`X`. Note that the produced complex includes one extra rank 1 module on each end, with the zero map leading to it, to simulate the maps to and from the zero module that would end a chain complex constructed from a simplicial complex in ordinary cases. .. magma:example:: Example: chaincomplex :label: chaincomplex .. code-block:: magma > ChainComplex(SimplicialComplex([{1}]),Integers()); Chain complex with terms of degree 2 down to -1 Dimensions of terms: 1 1 1 1 > BoundaryMaps(ChainComplex(SimplicialComplex([{1}]),Integers())); [* \ [0], \ [-1], \ [0] *] > ChainComplex(SimplicialComplex([{1,2,3}]),GF(3)); Chain complex with terms of degree 4 down to -1 Dimensions of terms: 1 1 3 3 1 1 > BoundaryMaps(ChainComplex(SimplicialComplex([{1,2,3}]),GF(3))); [* \ [0], \ [1 2 2], \ [1 0 2] \ [0 1 2] \ [1 2 0], \ [2] \ [2] \ [2], \ [0] *] .. magma:function:: HomologyGenerators(X) :input_types: SmpCpx :label: HomologyGenerators_SmpCpx .. magma:function:: HomologyGenerators(X, A) :input_types: SmpCpx, Rng :label: HomologyGenerators_SmpCpx_Rng .. magma:function:: HomologyGenerators(H, M, X) :input_types: SeqEnum, SeqEnum, SmpCpx :label: HomologyGenerators_SeqEnum_SeqEnum_SmpCpx Prints generators of the homology groups of the simplicial complex :math:`X` with coefficients in :math:`A` together with their order, in order of dimension. The latter calling form expects :math:`H,M` to be the result from ``H,M := ChainComplex(A,smp);``. This function will recalculate homology each time unless the homology is already cached in the simplicial complex using ``Homology(A, smp)``. If no ring is given, the function defaults to integer coefficients. .. magma:example:: Example: homologygenerators :label: homologygenerators This function gives a condensed form of the actual bases of the homology groups, as well as mappings back to an actual chain representative for each homology class. The six found generators are the generators of each of the contained torus homology groups. Notice that each generator is printed out with a prefix. This gives the order of the generator - so that for instance torsion elements of homology may be recognized. Thus, we see with the projective plane: We can further take this opportunity to verify the claims about ``Cone`` and ``Suspension`` with regard to the homology. .. code-block:: magma > HomologyGenerators(threetorus,Integers()); *** dimension 2 *** inf: { 21, 23, 27 } - { 4, 6, 16 } - { 20, 24, 26 } + { 20, 25, 26 } - { 14, 19, 25 } + { 12, 14, 25 } + { 3, 6, 11 } - { 10, 14, 19 } + { 19, 20, 27 } - { 3, 9, 11 } - { 10, 13, 20 } + { 10, 13, 19 } - { 10, 12, 14 } - { 12, 16, 19 } - { 11, 19, 20 } - { 13, 16, 20 } + { 21, 24, 25 } + { 24, 26, 27 } - { 20, 21, 24 } - { 19, 22, 26 } - { 3, 10, 16 } - { 4, 6, 10 } - { 22, 25, 26 } + { 11, 16, 20 } + { 4, 8, 16 } - { 5, 6, 9 } + { 4, 8, 10 } - { 11, 12, 19 } - { 3, 5, 9 } + { 11, 12, 25 } + { 8, 9, 10 } + { 20, 21, 27 } + { 3, 6, 16 } + { 10, 11, 25 } + { 19, 22, 24 } - { 9, 10, 11 } + { 3, 5, 10 } + { 10, 20, 25 } - { 8, 11, 16 } - { 19, 26, 27 } - { 19, 23, 24 } + { 10, 12, 16 } - { 19, 21, 23 } - { 6, 8, 9 } + { 13, 16, 19 } + { 19, 21, 25 } + { 23, 24, 27 } - { 22, 24, 25 } + { 5, 6, 10 } + { 6, 8, 11 } *** dimension 1 *** inf: -1*{ 10, 13 } + { 3, 5 } - { 8, 9 } - { 8, 16 } - { 19, 22 } + { 20, 26 } + { 9, 11 } - { 3, 16 } + { 5, 10 } + { 22, 26 } + { 11, 20 } + 2*{ 10, 20 } + { 13, 19 } - 2*{ 10, 11 } inf: { 3, 9 } - { 3, 5 } - { 9, 11 } - { 5, 10 } + { 10, 11 } inf: -1*{ 26, 27 } - { 11, 19 } + 2*{ 19, 22 } - { 20, 26 } - 2*{ 22, 26 } + { 11, 20 } + { 19, 21 } - { 21, 27 } inf: { 10, 13 } + { 11, 19 } - { 11, 20 } - { 10, 20 } - { 13, 19 } inf: -1*{ 11, 19 } + { 19, 22 } - { 20, 26 } - { 22, 26 } + { 11, 20 } inf: { 11, 20 } + { 10, 20 } - { 10, 11 } > HomologyGenerators(SimplicialProjectivePlane(),Integers()); *** dimension 1 *** 2: { 3, 6 } + { 2, 3 } - { 2, 6 } > HomologyGenerators(Cone(SimplicialProjectivePlane()),Integers()); Complex is acyclic. > HomologyGenerators(Suspension(SimplicialProjectivePlane()),Integers()); *** dimension 2 *** 2: { 1, 5, 7 } - { 1, 4, 7 } + { 1, 2, 5 } + { 1, 5, 8 } - { 2, 3, 8 } - { 3, 5, 7 } - { 1, 2, 8 } + { 5, 6, 8 } + { 1, 4, 5 } - { 4, 6, 8 } + { 3, 4, 7 } + { 2, 3, 5 } - { 4, 5, 6 } - { 3, 4, 8 }