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inducedReducedSimplicialChainComplexMap -- the induced maps that arise via inclusions of abstract simplicial complexes

Description

If an abstract simplicial complex can be regarded as a subsimplicial complex of another abstract simplicial complex, then it is useful to calculate the induced map at the level of reduced simplicial chain complexes. This is made possible by the method inducedReducedSimplicialChainComplexMap.

i1 : K = abstractSimplicialComplex({{1,2},{3}})

o1 = AbstractSimplicialComplex{-1 => {{}}          }
                               0 => {{1}, {2}, {3}}
                               1 => {{1, 2}}

o1 : AbstractSimplicialComplex
 
i2 : J = ambientAbstractSimplicialComplex(K)

o2 = AbstractSimplicialComplex{-1 => {{}}                   }
                               0 => {{1}, {2}, {3}}
                               1 => {{1, 2}, {1, 3}, {2, 3}}
                               2 => {{1, 2, 3}}

o2 : AbstractSimplicialComplex
 
i3 : inducedReducedSimplicialChainComplexMap(J,K)

            1              1
o3 = -1 : ZZ  <--------- ZZ  : -1
                 | 1 |

           3                  3
     0 : ZZ  <------------- ZZ  : 0
                | 1 0 0 |
                | 0 1 0 |
                | 0 0 1 |

           3              1
     1 : ZZ  <--------- ZZ  : 1
                | 1 |
                | 0 |
                | 0 |

o3 : ComplexMap
 
i4 : L = abstractSimplicialComplex {{}}

o4 = AbstractSimplicialComplex{-1 => {{}}}

o4 : AbstractSimplicialComplex
 
i5 : inducedReducedSimplicialChainComplexMap(L,L)

o5 = -2 : 0 <----- 0 : -2
               0

            1              1
     -1 : ZZ  <--------- ZZ  : -1
                 | 1 |

o5 : ComplexMap
 
i6 : M = abstractSimplicialComplex {{1}}

o6 = AbstractSimplicialComplex{-1 => {{}}}
                               0 => {{1}}

o6 : AbstractSimplicialComplex
 
i7 : L = abstractSimplicialComplex {{}}

o7 = AbstractSimplicialComplex{-1 => {{}}}

o7 : AbstractSimplicialComplex
 
i8 : inducedReducedSimplicialChainComplexMap(M,L)

o8 = -2 : 0 <----- 0 : -2
               0

            1              1
     -1 : ZZ  <--------- ZZ  : -1
                 | 1 |

o8 : ComplexMap

See also

Ways to use inducedReducedSimplicialChainComplexMap:

  • inducedReducedSimplicialChainComplexMap(AbstractSimplicialComplex,AbstractSimplicialComplex)

For the programmer

The object inducedReducedSimplicialChainComplexMap is a method function.

The source of this document is in AbstractSimplicialComplexes.m2:820:0.