edu.stanford.math.plex4.homology.chain_basis

Class Cell

java.lang.Object

edu.stanford.math.plex4.homology.chain_basis.Cell

All Implemented Interfaces:

PrimitiveBasisElement

public class Cell
extends java.lang.Object
implements PrimitiveBasisElement

This class implements the functionality of a cell within a CW complex. A CW complex can be described as follows:

1. Start with a discrete set X_0 consisting of the 0-cells
2. Inductively form the n-skeleton by attaching n-cells e_a via the maps phi_a: S^(n-1) -> X_{n-1} which map the boundary of e_a into the n-1 skeleton of X.

To define a cell within the CW complex, we require its dimensionality, its boundary, and the degrees of the map attaching map followed by a quotient map. The user of this class is advised to be familiar with cellular homology, as described in (for example) section 2.2 of the book "Algebraic Topology" by Allen Hatcher.

The required input degrees are the degrees of the composite map S_a^{n-1}

For example, we will construct a torus using a cell complex.

• Start with 1 0-cell, called v.
• Add 2 1-cells, a and b, each with boundary glued to the 0-cell. The degree of such a mapping is zero.
• Add a 2-cell with boundary elements (a, b, a, b), and degrees (1, -1, 1, -1)

This corresponds to the familiar construction of a square with opposite edges identified.

Author:

Andrew Tausz

Constructor Summary

Constructors

Constructor and Description
Cell() This constructor initializes the cell to be a 0-cell (a vertex), with the default cellId as the vertex index.
Cell(int dimension, Cell[] boundaryElements) This constructor initializes a cell to be an n-cell with n > 0.
Cell(int dimension, java.util.Collection<Cell> boundaryElements) This constructor initializes a cell to be an n-cell with n > 0.
Cell(int dimension, java.util.Collection<Cell> boundaryElements, int[] attachingDegrees) This constructor initializes the cell to be an n-cell with n > 0.

Method Summary

Methods

Modifier and Type	Method and Description
boolean	equals(java.lang.Object obj)
PrimitiveBasisElement[]	getBoundaryArray() This function returns the boundary of the current basis element.
int[]	getBoundaryCoefficients() This function returns the coefficients of the elements of the boundary array.
int[]	getBoundaryIds() This function returns the cell ids of the boundary elements.
int	getCellId() This function returns the unique identifier of the cell.
int	getDimension() This function returns the dimension of the basis element.
int	hashCode()
java.lang.String	toString()
void	verifyDimension() This function checks to see if the cell has been glued to boundary elements all of the same dimension.

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Constructor Detail

Cell

public Cell()

This constructor initializes the cell to be a 0-cell (a vertex), with the default cellId as the vertex index.

Cell

public Cell(int dimension,
    java.util.Collection<Cell> boundaryElements)

This constructor initializes a cell to be an n-cell with n > 0. Such a cell is specified by its boundary elements. The degrees of the attaching maps for this constructor are taken to be the default [1, -1, 1, -1, ...].

Parameters:

dimension - the geometric dimension of the cell

boundaryElements - an array containing the objects in the boundary

Cell

public Cell(int dimension,
    java.util.Collection<Cell> boundaryElements,
    int[] attachingDegrees)

This constructor initializes the cell to be an n-cell with n > 0. It allows the specification of the boundary elements as well as the attaching degrees.

Parameters:

dimension - the geometric dimension of the cell

boundaryElements - an array containing the objects in the boundary

attachingDegrees - the degrees of the attaching maps to the boundary objects

Cell

public Cell(int dimension,
    Cell[] boundaryElements)

This constructor initializes a cell to be an n-cell with n > 0. Such a cell is specified by its boundary elements. The degrees of the attaching maps for this constructor are taken to be the default [1, -1, 1, -1, ...].

Parameters:

dimension - the geometric dimension of the cell

boundaryElements - an array containing the objects in the boundary

Method Detail

getBoundaryIds

public int[] getBoundaryIds()

This function returns the cell ids of the boundary elements.

Returns:

an array containing the cell ids of the boundary elements.

getCellId

public int getCellId()

This function returns the unique identifier of the cell.

Returns:

the cellId of the cell

getBoundaryArray

public PrimitiveBasisElement[] getBoundaryArray()

Description copied from interface: PrimitiveBasisElement

This function returns the boundary of the current basis element. As with the dimension, this actually obtains the geometric boundary of this object. Note that this function only returns the the objects in the boundary, but not their coefficients.

Specified by:

getBoundaryArray in interface PrimitiveBasisElement

Returns:

the boundary of the current object

getDimension

public int getDimension()

Description copied from interface: PrimitiveBasisElement

This function returns the dimension of the basis element. Note that this is the actual geometric dimension of the simplex or cell. For example, for a simplex this would be one less than the number of indices in it.

Specified by:

getDimension in interface PrimitiveBasisElement

Returns:

the geometric dimension of this object

getBoundaryCoefficients

public int[] getBoundaryCoefficients()

Description copied from interface: PrimitiveBasisElement

This function returns the coefficients of the elements of the boundary array. It is implemented as a separate function, since this interface is designed to be agnostic to its algebraic environment. In the case of the Simplex class this will return the array consisting of alternating signs [1, -1, 1, ...]. However, in the case of the Cell class it will return the degrees of the composite attaching maps used to compute cellular homology.

Specified by:

getBoundaryCoefficients in interface PrimitiveBasisElement

Returns:

the coefficients of the boundary elements

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object

hashCode

public int hashCode()

Overrides:

hashCode in class java.lang.Object

equals

public boolean equals(java.lang.Object obj)

Overrides:

equals in class java.lang.Object

verifyDimension

public void verifyDimension()

This function checks to see if the cell has been glued to boundary elements all of the same dimension.