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/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2008, Ben Burton *
* For further details contact Ben Burton (bab@debian.org). *
* *
* This program is free software; you can redistribute it and/or *
* modify it under the terms of the GNU General Public License as *
* published by the Free Software Foundation; either version 2 of the *
* License, or (at your option) any later version. *
* *
* This program is distributed in the hope that it will be useful, but *
* WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU *
* General Public License for more details. *
* *
* You should have received a copy of the GNU General Public *
* License along with this program; if not, write to the Free *
* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, *
* MA 02110-1301, USA. *
* *
**************************************************************************/
/* end stub */
// ntriangulationhomology.cpp
// Implements homology as a property of a triangulation.
#include "triangulation/ntriangulation.h"
#include "maths/nmatrixint.h"
namespace regina {
const NAbelianGroup& NTriangulation::getHomologyH1() const {
if (H1.known())
return *H1.value();
if (getNumberOfTetrahedra() == 0)
return *(H1 = new NAbelianGroup());
// Calculate the first homology.
// Find a maximal forest in the dual 1-skeleton.
// Note that this will ensure the skeleton has been calculated.
stdhash::hash_set<NFace*, HashPointer> forest;
maximalForestInDualSkeleton(forest);
// Build a presentation matrix.
// Each non-boundary not-in-forest face is a generator.
// Each non-boundary edge is a relation.
unsigned long nBdryEdges = 0;
unsigned long nBdryFaces = 0;
for (BoundaryComponentIterator bit = boundaryComponents.begin();
bit != boundaryComponents.end(); bit++) {
nBdryEdges += (*bit)->getNumberOfEdges();
nBdryFaces += (*bit)->getNumberOfFaces();
}
long nGens = getNumberOfFaces() - nBdryFaces - forest.size();
long nRels = getNumberOfEdges() - nBdryEdges;
NMatrixInt pres(nRels, nGens);
// Find out which face corresponds to which generator.
long* genIndex = new long[getNumberOfFaces()];
long i = 0;
for (FaceIterator fit = faces.begin(); fit != faces.end(); fit++) {
if ((*fit)->isBoundary())
genIndex[fit - faces.begin()] = -1;
else if (forest.count(*fit))
genIndex[fit - faces.begin()] = -1;
else {
genIndex[fit - faces.begin()] = i;
i++;
}
}
// Run through each edge and put the relations in the matrix.
std::deque<NEdgeEmbedding>::const_iterator embit;
NTetrahedron* currTet;
NFace* face;
int currTetFace;
long faceGenIndex;
i = 0;
for (EdgeIterator eit = edges.begin(); eit != edges.end(); eit++) {
if (! (*eit)->isBoundary()) {
// Put in the relation corresponding to this edge.
for (embit = (*eit)->getEmbeddings().begin();
embit != (*eit)->getEmbeddings().end(); embit++) {
currTet = (*embit).getTetrahedron();
currTetFace = (*embit).getVertices()[2];
face = currTet->getFace(currTetFace);
faceGenIndex = genIndex[faceIndex(face)];
if (faceGenIndex >= 0) {
if ((face->getEmbedding(0).getTetrahedron() == currTet) &&
(face->getEmbedding(0).getFace() == currTetFace))
pres.entry(i, faceGenIndex) += 1;
else
pres.entry(i, faceGenIndex) -= 1;
}
}
i++;
}
}
delete[] genIndex;
// Build the group from the presentation matrix and tidy up.
NAbelianGroup* ans = new NAbelianGroup();
ans->addGroup(pres);
return *(H1 = ans);
}
const NAbelianGroup& NTriangulation::getHomologyH1Rel() const {
if (H1Rel.known())
return *H1Rel.value();
if (getNumberOfBoundaryComponents() == 0)
return *(H1Rel = new NAbelianGroup(getHomologyH1()));
// Calculate the relative first homology wrt the boundary.
// Find a maximal forest in the 1-skeleton.
// Note that this will ensure the skeleton has been calculated.
stdhash::hash_set<NEdge*, HashPointer> forest;
maximalForestInSkeleton(forest, false);
// Build a presentation matrix.
// Each non-boundary not-in-forest edge is a generator.
// Each non-boundary face is a relation.
unsigned long nBdryVertices = 0;
unsigned long nBdryEdges = 0;
unsigned long nBdryFaces = 0;
unsigned long nClosedComponents = 0;
for (BoundaryComponentIterator bit = boundaryComponents.begin();
bit != boundaryComponents.end(); bit++) {
nBdryVertices += (*bit)->getNumberOfVertices();
nBdryEdges += (*bit)->getNumberOfEdges();
nBdryFaces += (*bit)->getNumberOfFaces();
}
for (ComponentIterator cit = components.begin();
cit != components.end(); cit++)
if ((*cit)->isClosed())
nClosedComponents++;
long nGens = getNumberOfEdges() - nBdryEdges
- getNumberOfVertices() + nBdryVertices
+ nClosedComponents;
long nRels = getNumberOfFaces() - nBdryFaces;
NMatrixInt pres(nRels, nGens);
// Find out which edge corresponds to which generator.
long* genIndex = new long[getNumberOfEdges()];
long i = 0;
for (EdgeIterator eit = edges.begin(); eit != edges.end(); eit++) {
if ((*eit)->isBoundary())
genIndex[eit - edges.begin()] = -1;
else if (forest.count(*eit))
genIndex[eit - edges.begin()] = -1;
else {
genIndex[eit - edges.begin()] = i;
i++;
}
}
// Run through each face and put the relations in the matrix.
NTetrahedron* currTet;
NPerm currTetVertices;
long edgeGenIndex;
i = 0;
int faceEdge, currEdgeStart, currEdgeEnd, currEdge;
for (FaceIterator fit = faces.begin(); fit != faces.end(); fit++) {
if (! (*fit)->isBoundary()) {
// Put in the relation corresponding to this face.
currTet = (*fit)->getEmbedding(0).getTetrahedron();
currTetVertices = (*fit)->getEmbedding(0).getVertices();
for (faceEdge = 0; faceEdge < 3; faceEdge++) {
currEdgeStart = currTetVertices[faceEdge];
currEdgeEnd = currTetVertices[(faceEdge + 1) % 3];
// Examine the edge from vertex edgeStart to edgeEnd
// in tetrahedron currTet.
currEdge = edgeNumber[currEdgeStart][currEdgeEnd];
edgeGenIndex = genIndex[edgeIndex(
currTet->getEdge(currEdge))];
if (edgeGenIndex >= 0) {
if (currTet->getEdgeMapping(currEdge)[0] == currEdgeStart)
pres.entry(i, edgeGenIndex) += 1;
else
pres.entry(i, edgeGenIndex) -= 1;
}
}
i++;
}
}
delete[] genIndex;
// Build the group from the presentation matrix and tidy up.
NAbelianGroup* ans = new NAbelianGroup();
ans->addGroup(pres);
return *(H1Rel = ans);
}
const NAbelianGroup& NTriangulation::getHomologyH1Bdry() const {
if (H1Bdry.known())
return *H1Bdry.value();
// Run through the individual boundary components and add the
// appropriate pieces to the homology group.
unsigned long rank = 0;
unsigned long z2rank = 0;
// Ensure that the skeleton has been calculated.
if (! calculatedSkeleton)
calculateSkeleton();
for (BoundaryComponentIterator bit = boundaryComponents.begin();
bit != boundaryComponents.end(); bit++) {
if ((*bit)->isOrientable()) {
rank += (2 - (*bit)->getEulerCharacteristic());
} else {
rank += (1 - (*bit)->getEulerCharacteristic());
z2rank++;
}
}
// Build the group and tidy up.
NAbelianGroup* ans = new NAbelianGroup();
ans->addRank(rank);
ans->addTorsionElement(2, z2rank);
return *(H1Bdry = ans);
}
const NAbelianGroup& NTriangulation::getHomologyH2() const {
if (H2.known())
return *H2.value();
if (getNumberOfTetrahedra() == 0)
return *(H2 = new NAbelianGroup());
// Calculations are different for orientable vs non-orientable
// components.
// We know the only components will be Z and Z_2.
long rank, z2rank;
if (isOrientable()) {
// Same as H1Rel without the torsion elements.
rank = getHomologyH1Rel().getRank();
z2rank = 0;
} else {
// Non-orientable!
// z2rank = # closed cmpts - # closed orientable cmpts
z2rank = 0;
for (ComponentIterator cit = components.begin();
cit != components.end(); cit++)
if ((*cit)->isClosed())
if (! ((*cit)->isOrientable()))
z2rank++;
// Find rank(Z_2) + rank(Z) and take off z2rank.
rank = getHomologyH1Rel().getRank() +
getHomologyH1Rel().getTorsionRank(2) -
getHomologyH1().getTorsionRank(2) -
z2rank;
}
// Build the new group and tidy up.
NAbelianGroup* ans = new NAbelianGroup();
ans->addRank(rank);
if (z2rank)
ans->addTorsionElement(2, z2rank);
return *(H2 = ans);
}
} // namespace regina
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