Macaulay2 » Documentation
Packages » AbstractSimplicialComplexes :: inducedSimplicialChainComplexMap
next | previous | forward | backward | up | index | toc

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

- Usage:
    `inducedSimplicialChainComplexMap(K,L)`
- Inputs:
  - K, an abstract simplicial complex,
  - L, an abstract simplicial complex,
- Outputs:
  - a map of 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 simplicial chain complexes. This is made possible by the method inducedSimplicialChainComplexMap.

    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 : inducedSimplicialChainComplexMap(J,K)

               3                  3
    o3 = 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 : inducedSimplicialChainComplexMap(L,L)

    o5 = 0

    o5 : ComplexMap

    i6 : M = abstractSimplicialComplex {{1}}

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

    o6 : AbstractSimplicialComplex

    i7 : L = abstractSimplicialComplex {{}}

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

    o7 : AbstractSimplicialComplex

    i8 : inducedSimplicialChainComplexMap(M,L)

    o8 = 0

    o8 : ComplexMap

## See also

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

## Ways to use inducedSimplicialChainComplexMap:

- `inducedSimplicialChainComplexMap(AbstractSimplicialComplex,AbstractSimplicialComplex)`

## For the programmer

The object inducedSimplicialChainComplexMap is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2:790:0.