Regina Calculation Engine
Public Member Functions
regina::NHomMarkedAbelianGroup Class Reference

Represents a homomorphism of finitely generated abelian groups. More...

#include <algebra/nmarkedabeliangroup.h>

Inheritance diagram for regina::NHomMarkedAbelianGroup:
regina::ShareableObject regina::boost::noncopyable

List of all members.

Public Member Functions

 NHomMarkedAbelianGroup (const NMarkedAbelianGroup &dom, const NMarkedAbelianGroup &ran, const NMatrixInt &mat)
 Constructs a homomorphism from two marked abelian groups and a matrix that indicates where the generators are sent.
 NHomMarkedAbelianGroup (const NHomMarkedAbelianGroup &h)
 Copy constructor.
 ~NHomMarkedAbelianGroup ()
 Destructor.
bool isChainMap (const NHomMarkedAbelianGroup &other) const
 Determines whether this and the given homomorphism together form a chain map.
bool isCycleMap () const
 Is this at least a cycle map? If not, pretty much any further computations you try with this class will be give you nothing more than carefully-crafted garbage.
bool isEpic () const
 Is this an epic homomorphism?
bool isMonic () const
 Is this a monic homomorphism?
bool isIso () const
 A deprecated alternative to isIsomorphism().
bool isIsomorphism () const
 Is this an isomorphism?
bool isZero () const
 Is this the zero map?
bool isIdentity () const
 Is this the identity automorphism?
const NMarkedAbelianGroupgetKernel () const
 Returns the kernel of this homomorphism.
const NMarkedAbelianGroupgetCokernel () const
 Returns the cokernel of this homomorphism.
const NMarkedAbelianGroupgetImage () const
 Returns the image of this homomorphism.
void writeTextShort (std::ostream &out) const
 Short text representation.
void writeTextLong (std::ostream &out) const
 A more detailed text representation of the homomorphism.
const NMarkedAbelianGroupgetDomain () const
 Returns the domain of this homomorphism.
const NMarkedAbelianGroupgetRange () const
 Returns the range of this homomorphism.
const NMatrixIntgetDefiningMatrix () const
 Returns the defining matrix for the homomorphism.
const NMatrixIntgetReducedMatrix () const
 Returns the internal reduced matrix representing the homomorphism.
std::vector< NLargeIntegerevalCC (const std::vector< NLargeInteger > &input) const
 Evaluate the image of a vector under this homomorphism, using the original chain complexes' coordinates.
std::vector< NLargeIntegerevalSNF (const std::vector< NLargeInteger > &input) const
 Evaluate the image of a vector under this homomorphism, using the Smith normal form coordinates.
std::auto_ptr
< NHomMarkedAbelianGroup
inverseHom () const
 Returns the inverse to an NHomMarkedAbelianGroup.
std::auto_ptr
< NHomMarkedAbelianGroup
operator* (const NHomMarkedAbelianGroup &X) const
 Returns the composition of two homomorphisms.
void writeReducedMatrix (std::ostream &out) const
 Writes a human-readable version of the reduced matrix to the given output stream.

Detailed Description

Represents a homomorphism of finitely generated abelian groups.

One initializes such a homomorphism by providing:

So for example, if the domain was initialized by the chain complex Z^a --A--> Z^b --B--> Z^c with mod p coefficients, and the range was initialized by Z^d --D--> Z^e --E--> Z^f with mod q coefficients, then the matrix needs to be an e-by-b matrix. Furthermore, you only obtain a well-defined homomorphism if this matrix extends to a cycle map, which this class assumes but which the user can confirm with isCycleMap(). Moreover, q should divide p: this allows for q > 0 and p = 0, which means the domain has Z coefficients and the range has mod q coefficients.

Todo:
Optimise (long-term): preImageOf in CC and SNF coordinates. This routine would return a generating list of elements in the preimage, thought of as an affine subspace. Or maybe just one element together with the kernel inclusion. IMO smarter to be a list because that way there's a more pleasant way to make it empty. Or we could have a variety of routines among these themes. Store some minimal data for efficient computations of preImage, eventually replacing the internals of inverseHom() with a more flexible set of tools. Also add an isInImage() in various coordinates.
Todo:
Optimise (long-term): writeTextShort() have completely different set of descriptors if an endomorphism domain = range (not so important at the moment though). New descriptors would include things like automorphism, projection, differential, finite order, etc.
Todo:
Optimise (long-term): Add map factorization, so that every homomorphism can be split as a composite of a projection followed by an inclusion. Add kernelInclusion(), coKerMap(), etc. Add a liftMap() call, i.e., a procedure to find a lift of a map if one exists.
Author:
Ryan Budney

Constructor & Destructor Documentation

Constructs a homomorphism from two marked abelian groups and a matrix that indicates where the generators are sent.

The roles of the two groups and the matrix are described in detail in the NHomMarkedAbelianGroup class overview.

The matrix must be given in the chain-complex coordinates. Specifically, if the domain was defined via the chain complex Z^a --N1--> Z^b --M1--> Z^c and the range was defined via Z^d --N2--> Z^e --M2--> Z^f, then mat is an e-by-b matrix that describes a homomorphism from Z^b to Z^e.

In order for this to make sense as a homomorphism of the groups represented by the domain and range respectively, one requires img(mat*N1) to be a subset of img(N2). Similarly, ker(M1) must be sent into ker(M2). These facts are not checked, but are assumed as preconditions of this constructor.

Precondition:
The matrix mat has the required dimensions e-by-b, gives img(mat*N1) as a subset of img(N2), and sends ker(M1) into ker(M2), as explained in the detailed notes above.
Parameters:
domthe domain group.
ranthe range group.
matthe matrix that describes the homomorphism from dom to ran.

Copy constructor.

Parameters:
hthe homomorphism to clone.

Destructor.


Member Function Documentation

std::vector<NLargeInteger> regina::NHomMarkedAbelianGroup::evalCC ( const std::vector< NLargeInteger > &  input) const

Evaluate the image of a vector under this homomorphism, using the original chain complexes' coordinates.

This involves multiplication by the defining matrix.

Python:
Not available yet. This routine will be made accessible to Python in a future release.
Parameters:
inputan input vector in the domain chain complex's coordinates, of length getDomain().getM().columns().
Returns:
the image of this vector in the range chain complex's coordinates, of length getRange().getM().columns().
std::vector<NLargeInteger> regina::NHomMarkedAbelianGroup::evalSNF ( const std::vector< NLargeInteger > &  input) const

Evaluate the image of a vector under this homomorphism, using the Smith normal form coordinates.

This is just multiplication by the reduced matrix, returning the empty vector if the input vector has the wrong dimensions.

Warning:
Smith normal form coordinates are sensitive to the implementation of the Smith Normal Form, i.e., they are not canonical.
Python:
Not available yet. This routine will be made accessible to Python in a future release.
Parameters:
inputan input vector in the domain SNF coordinates, of length getDomain().minNumberOfGenerators().
Returns:
the image of this vector in the range chain complex's coordinates, of length getRange().minNumberOfGenerators().

Returns the cokernel of this homomorphism.

Returns:
the cokernel of the homomorphism, as a marked abelian group.

Returns the defining matrix for the homomorphism.

Returns:
the matrix that was used to define the homomorphism.

Returns the domain of this homomorphism.

Returns:
the domain that was used to define the homomorphism.

Returns the image of this homomorphism.

Returns:
the image of the homomorphism, as a marked abelian group.

Returns the kernel of this homomorphism.

Returns:
the kernel of the homomorphism, as a marked abelian group.

Returns the range of this homomorphism.

Returns:
the range that was used to define the homomorphism.

Returns the internal reduced matrix representing the homomorphism.

This is where the rows/columns of the matrix represent first the free generators, then the torsion summands in the order of the invariant factors:

Z^d + Z_{d0} + ... + Z_{dk} where:

  • d is the number of free generators, as returned by getRank();
  • d1, ..., dk are the invariant factors that describe the torsion elements of the group, where 1 < d1 | d2 | ... | dk.
Returns:
a copy of the internal representation of the homomorphism.

Returns the inverse to an NHomMarkedAbelianGroup.

If this homomorphism is not invertible, this routine returns the zero homomorphism.

If you are computing with mod-p coefficients, this routine will further require that this invertible map preserves the UCT splitting of the group, i.e., it gives an isomorphism of the tensor product parts and the TOR parts. At present this suffices since we're only using this to construct maps between homology groups in different coordinate systems.

Returns:
the inverse homomorphism, or the zero homomorphism if this is not invertible.

Determines whether this and the given homomorphism together form a chain map.

Given two NHomMarkedAbelianGroups, you have two diagrams:

 Z^a --N1--> Z^b --M1--> Z^c   Z^g --N3--> Z^h --M3--> Z^i
                   ^                             ^
                   |this.matrix                  |other.matrix
 Z^d --N2--> Z^e --M2--> Z^f   Z^j --N4--> Z^k --M4--> Z^l
 

If c=g and f=j and M1=N3 and M2=N4, you can ask if these maps commute, i.e., whether you have a map of chain complexes.

Parameters:
otherthe other homomorphism to analyse in conjunction with this.
Returns:
true if and only if c=g, M1=N3, f=j, M2=N4, and the diagram commutes.

Is this at least a cycle map? If not, pretty much any further computations you try with this class will be give you nothing more than carefully-crafted garbage.

Technically, this routine only checks that cycles are sent to cycles, since it only has access to three of the four maps you need to verify you have a cycle map.

Returns:
true if and only if this is a chain map.
bool regina::NHomMarkedAbelianGroup::isEpic ( ) const [inline]

Is this an epic homomorphism?

Returns:
true if this homomorphism is epic.

Is this the identity automorphism?

Returns:
true if and only if the domain and range are defined via the same chain complexes and the induced map on homology is the identity.
bool regina::NHomMarkedAbelianGroup::isIso ( ) const [inline]

A deprecated alternative to isIsomorphism().

Deprecated:
This routine will be removed in a future version of Regina; please use the identical routine isIsomorphism() instead.
Returns:
true if this homomorphism is an isomorphism.

Is this an isomorphism?

Returns:
true if this homomorphism is an isomorphism.
bool regina::NHomMarkedAbelianGroup::isMonic ( ) const [inline]

Is this a monic homomorphism?

Returns:
true if this homomorphism is monic.
bool regina::NHomMarkedAbelianGroup::isZero ( ) const [inline]

Is this the zero map?

Returns:
true if this homomorphism is the zero map.
std::auto_ptr<NHomMarkedAbelianGroup> regina::NHomMarkedAbelianGroup::operator* ( const NHomMarkedAbelianGroup X) const

Returns the composition of two homomorphisms.

Precondition:
the homomorphisms must be composable, meaning that the range of X must have the same presentation matrices as the domain of this homomorphism.
Parameters:
Xthe homomorphism to compose this with.
Returns:
a newly created composite homomorphism.
void regina::NHomMarkedAbelianGroup::writeReducedMatrix ( std::ostream &  out) const

Writes a human-readable version of the reduced matrix to the given output stream.

This is a description of the homomorphism in some specific coordinates at present only meant to be internal to NHomMarkedAbelianGroup. At present, these coordinates have the torsion factors of the group appearing first, followed by the free factors.

Python:
The out argument is missing; instead this is assumed to be standard output.
Parameters:
outthe output stream.
void regina::NHomMarkedAbelianGroup::writeTextLong ( std::ostream &  out) const [virtual]

A more detailed text representation of the homomorphism.

Parameters:
outthe stream to write to.

Reimplemented from regina::ShareableObject.

void regina::NHomMarkedAbelianGroup::writeTextShort ( std::ostream &  out) const [virtual]

Short text representation.

This will state some basic properties of the homomorphism, such as:

  • whether the map is the identity;
  • whether the map is an isomorphism;
  • whether the map is monic or epic;
  • if it is not monic, describes the kernel;
  • if it is not epic, describes the co-kernel;
  • if it is neither monic nor epic, describes the image.
Parameters:
outthe stream to write to.

Implements regina::ShareableObject.


The documentation for this class was generated from the following file:

Copyright © 1999-2011, The Regina development team
This software is released under the GNU General Public License.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@debian.org).