Definition

The concept AbstractChainComplex describes the requirements for (topological) chain complexes associated to abstract complexes used in the concept HDVF.

It provides methods to:

get the dimension of the complex, the number of cells in each dimension
get the boundary and co-boundary of cell(s)
get the vertices of a given cell
output the complex in text format

Cells are indexed along each dimension and thus identified by their index together with their dimension.

Has models
: CGAL::Homological_discrete_vector_field::Abstract_simplicial_chain_complex<IntegralDomainWithoutDivision>
: CGAL::Homological_discrete_vector_field::Sub_chain_complex_mask<IntegralDomainWithoutDivision, AbstractChainComplex>

Related Functions
(Note that these are not member functions.)
std::ostream &	operator<< (std::ostream &out, const AbstractChainComplex &complex) const
	Inserts the chain complex in text mode in the stream.

Types
typedef CoefficientRing	Coefficient_ring
	Type of coefficients ring used to compute homology, model of `IntegralDomainWithoutDivision`

typedef CGAL::OSM::Sparse_chain< CoefficientRing, CGAL::OSM::COLUMN >	Column_chain
	Type of column-major chains (returned by the boundary operator)

typedef CGAL::OSM::Sparse_chain< CoefficientRing, CGAL::OSM::ROW >	Row_chain
	Type of row-major chains (returned by the co-boundary operator)

typedef CGAL::OSM::Sparse_matrix< CoefficientRing, CGAL::OSM::COLUMN >	Column_matrix
	Type of column-major sparse matrices (used to store the boundary operator)

Operators
AbstractChainComplex &	operator= (const AbstractChainComplex &complex)
	Assignment operator.

Access functions
int	dimension ()
	Returns the dimension of the complex, that is, the largest dimension of cells.

size_t	number_of_cells (int q)
	Returns the number of cells of dimension `q`.

const vector< Column_matrix > &	boundary_matrices () const
	Returns all boundary matrices.

const Column_matrix &	boundary_matrix (int q) const
	Returns the boundary matrix of dimension `q` (ie. the matrix of \(\partial_q\)).

Column_chain	d (size_t id_cell, int q)
	Returns the boundary of the cell of index `id_cell` in dimension `q`.

Row_chain	cod (size_t id_cell, int q)
	Returns the co-boundary of the cell of index `id_cell` in dimension `q`.

std::vector< size_t >	bottom_faces (size_t id_cell, int q) const
	Returns the vertices of a given cell (that is, the indices of its faces of dimension 0).

template<typename CoefficientR , int StorageF>
Column_chain	cofaces_chain (SparseChain< CoefficientR, StorageF > chain, int q) const
	Returns the cofaces of a given chain in dimension `q`.

Member Typedef Documentation

◆ Coefficient_ring

typedef CoefficientRing AbstractChainComplex::Coefficient_ring

Type of coefficients ring used to compute homology, model of IntegralDomainWithoutDivision

Member Function Documentation

◆ boundary_matrices()

const vector< Column_matrix > & AbstractChainComplex::boundary_matrices ( ) const

Returns all boundary matrices.

The function returns constant reference to a vector of column-major sparse matrices. The q-th element of this vector is the matrix of \(\partial_q\), which gives the boundary of cells of dimension q(as a linear combination of q-1 cells).

◆ boundary_matrix()

const Column_matrix & AbstractChainComplex::boundary_matrix ( int q ) const

Returns the boundary matrix of dimension q (ie. the matrix of \(\partial_q\)).

The function returns a column-major sparse matrices.

◆ cod()

Row_chain AbstractChainComplex::cod	(	size_t	id_cell,
		int	q
	)

Returns the co-boundary of the cell of index id_cell in dimension q.

This boundary is a finite linear combination of cells of dimension q+1. It is encoded as a row-major chain (which maps each cell with a non-zero coefficient to this coefficient). This co-boundary is thus the id_cell-th row of the boundary matrix in dimension q+1.

◆ cofaces_chain()

template<typename CoefficientR , int StorageF>

Column_chain AbstractChainComplex::cofaces_chain	(	SparseChain< CoefficientR, StorageF >	chain,
		int	q
	)		const

Returns the cofaces of a given chain in dimension q.

The resulting chain, whatever the storage format of the input, is column-major, lies in dimension q+1 and is null if this dimension exceeds the dimension of the complex.

Template Parameters

CoefficientR	`CoefficientRing` of the chain.
StorageF	`StorageFormat` of the chain.

◆ d()

Column_chain AbstractChainComplex::d	(	size_t	id_cell,
		int	q
	)

Returns the boundary of the cell of index id_cell in dimension q.

This boundary is a finite linear combination of cells of dimension q-1. It is encoded as a column-major chain (which maps each cell with a non-zero coefficient to this coefficient). This boundary is thus the id_cell-th column of the boundary matrix in dimension q.

◆ number_of_cells()

size_t AbstractChainComplex::number_of_cells ( int q )

Returns the number of cells of dimension q.

If q is negative of larger than the dimension of the complex, returns 0.

◆ operator=()

AbstractChainComplex & AbstractChainComplex::operator= ( const AbstractChainComplex & complex )

Assignment operator.

The operator creates a copy of complex.