Offers routines for constructing a variety of sample 4-dimensional triangulations. More...

#include <triangulation/example4.h>

Inheritance diagram for regina::Example< 4 >:

Static Public Member Functions
static Triangulation< dim >	sphere ()
	Closed Triangulations. More...

static Triangulation< dim >	simplicialSphere ()
	Returns the standard (dim+2)-simplex triangulation of the dim-sphere as the boundary of a (dim+1)-simplex. More...

static Triangulation< dim >	sphereBundle ()
	Returns a two-simplex triangulation of the product space `S^(dim-1) x S¹`. More...

static Triangulation< dim >	twistedSphereBundle ()
	Returns a two-simplex triangulation of the twisted product space `S^(dim-1) x~ S¹`. More...

static Triangulation< dim >	ball ()
	Bounded Triangulations. More...

static Triangulation< dim >	ballBundle ()
	Returns a triangulation of the product space `B^(dim-1) x S¹`. More...

static Triangulation< dim >	twistedBallBundle ()
	Returns a triangulation of the twisted product space `B^(dim-1) x~ S¹`. More...

static Triangulation< dim >	doubleCone (const Triangulation< dim-1 > &base)
	Returns a double cone over the given (dim-1)-dimensional triangulation. More...

static Triangulation< dim >	singleCone (const Triangulation< dim-1 > &base)
	Returns a single cone over the given (dim-1)-dimensional triangulation. More...

Closed Triangulations
static Triangulation< 4 >	fourSphere ()
	Returns a two-pentachoron triangulation of the 4-sphere. More...

static Triangulation< 4 >	simplicialFourSphere ()
	Returns the standard six-pentachoron triangulation of the 4-sphere as the boundary of a 5-simplex. More...

static Triangulation< 4 >	rp4 ()
	Returns a four-pentachoron triangulation of real projective 4-space. More...

static Triangulation< 4 >	cp2 ()
	Returns a four-pentachoron triangulation of the standard complex projective plane. More...

static Triangulation< 4 >	s2xs2 ()
	Returns a six-pentachoron triangulation of the standard product `S² x S²`. More...

static Triangulation< 4 >	s2xs2Twisted ()
	Returns a six-pentachoron triangulation of the twisted product `S² x~ S²`. More...

static Triangulation< 4 >	s3xs1 ()
	Returns a two-pentachoron triangulation of the product space `S³ x S¹`. More...

static Triangulation< 4 >	s3xs1Twisted ()
	Returns a two-pentachoron triangulation of the twisted product space `S³ x~ S¹`. More...

static Triangulation< 4 >	k3 ()
	Returns a triangulation of the standard K3 surface. More...

Ideal Triangulations
(end: Closed Triangulations)
static Triangulation< 4 >	cappellShaneson ()
	Returns a two-pentachoron triangulation of a Cappell-Shaneson 2-knot complement in the 4-sphere. More...

Constructions from 3-Manifold Triangulations
(end: Ideal Triangulations)
static Triangulation< 4 >	iBundle (const Triangulation< 3 > &base)
	Returns a triangulation of the product `M x I`, where M is the given 3-manifold triangulation. More...

static Triangulation< 4 >	s1Bundle (const Triangulation< 3 > &base)
	Returns a triangulation of the product `M x S1`, where M is the given 3-manifold triangulation. More...

static Triangulation< 4 >	bundleWithMonodromy (const Triangulation< 3 > &base, const Isomorphism< 3 > &monodromy)
	Returns a bundle formed from a given 3-manifold and a given monodromy. More...

Detailed Description

Offers routines for constructing a variety of sample 4-dimensional triangulations.

This is a specialisation of the generic Example class template; see the Example template documentation for a general overview of how the example triangulation classes work.

This 4-dimensional specialisation offers significant extra functionality, by providing several more hard-coded and parameterised constructions.

Member Function Documentation

◆ ball()

Triangulation< dim > regina::detail::ExampleBase< dim >::ball

staticinherited

Bounded Triangulations.

Returns a one-simplex triangulation of the dim-ball.

Returns: a one-simplex dim-ball.

◆ ballBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::ballBundle

staticinherited

Returns a triangulation of the product space B^(dim-1) x S¹.

This will use one simplex in odd dimensions, or two simplices in even dimensions.

Returns: the product B^(dim-1) x S¹.

◆ bundleWithMonodromy()

static Triangulation< 4 > regina::Example< 4 >::bundleWithMonodromy	(	const Triangulation< 3 > &	base,
		const Isomorphism< 3 > &	monodromy
	)

static

Returns a bundle formed from a given 3-manifold and a given monodromy.

Specifically, let M be the given 3-manifold triangulation. This routine builds the bundle M x I, and then identifies the two copies of M on the boundary according to the given homeomorphism from M to itself. The homeomorphism must be expressed as a combinatorial automorphism, which means that for a non-trivial monodromy you may need to do some work to find a sufficiently symmetric 3-manifold triangulation to begin with.

The resulting manifold will contain 82 pentachora for each original tetrahedron of M, and will contain many internal vertices. It is highly recommended that you call Triangulation<4>::intelligentSimplify() afterwards if you do not need to preserve the combinatorial structure.

Precondition: The given monodromy must be an isomorphism from M to itself; that is, a combinatorial automorphism.

Warning: If the given 3-manifold triangulation has ideal boundary, then you will obtain an invalid 4-manifold triangulation as a result.

Parameters

base	the 3-manifold triangulation M, as described above.
monodromy	the homeomorphism from M to itself, as described above.

Returns: the requested bundle.

◆ cappellShaneson()

static Triangulation< 4 > regina::Example< 4 >::cappellShaneson ( )

static

Returns a two-pentachoron triangulation of a Cappell-Shaneson 2-knot complement in the 4-sphere.

This triangulation is described and analysed in "Triangulating a Cappell-Shaneson knot complement", Budney, Burton and Hillman, Mathematical Research Letters 19 (2012), no. 5, 1117-1126.

Returns: a Cappell-Shaneson 2-knot complement.

◆ cp2()

static Triangulation< 4 > regina::Example< 4 >::cp2 ( )

static

Returns a four-pentachoron triangulation of the standard complex projective plane.

This triangulation is minimal.

Under the orientation convention that we use for intersection forms, this triangulation gives the "plain" CP² with intersection form [1], not the reflected CP² with intersection form [-1].

Returns: the standard complex projective plane.

◆ doubleCone()

Triangulation< dim > regina::detail::ExampleFromLowDim< dim, available >::doubleCone ( const Triangulation< dim-1 > & base )

staticinherited

Returns a double cone over the given (dim-1)-dimensional triangulation.

If the given triangulation represents the manifold M, then this returns an ideal triangulation of the product M x I (with two ideal boundary components). A copy of the original triangulation base can be found at the centre of this construction, formed from the dim-simplices that sit between the two ideal vertices.

Note that, as a special case, if M is either a sphere or a ball, then this routine returns a (dim)-sphere or a (dim)-ball (since "ideal spheres" and "ideal balls" just become regular internal and boundary vertices respectively).

This construction is essentially the suspension of the triangulation base. We do not call it this however, since from a topological point of view, to form the ideal triangulation of M x I we "remove" the vertices at the apex of each cone.

Warning: If the given (dim-1)-dimensional triangulation has any boundary whatsoever (either real or ideal), then unless it is a (dim-1)-ball, you will obtain an invalid dim-manifold triangulation as a result.

Returns: a double cone over the given triangulation.

◆ fourSphere()

Triangulation< 4 > regina::Example< 4 >::fourSphere ( )

inlinestatic

Returns a two-pentachoron triangulation of the 4-sphere.

This is identical to calling the generic routine sphere().

Returns: a two-pentachoron 4-sphere.

◆ iBundle()

static Triangulation< 4 > regina::Example< 4 >::iBundle ( const Triangulation< 3 > & base )

static

Returns a triangulation of the product M x I, where M is the given 3-manifold triangulation.

The boundary of this product will consist of two copies of M, both combinatorially isomorphic to the original triangulation. If n is the number of tetrahedra in M, then the first copy of M on the boundary is obtained by mapping vertices 0,1,2,3 of tetrahedron i of M to vertices 0,1,2,3 of pentachoron i, and the second copy is obtained by mapping vertices 0,1,2,3 of tetrahedron i of M to vertices 0,1,2,3 of pentachoron n+i.

The product itself will contain 82 pentachora for each original tetrahedron of M, and will contain many internal vertices. It is highly recommended that you call Triangulation<4>::intelligentSimplify() afterwards if you do not need to preserve the combinatorial structure.

Warning: If the given 3-manifold triangulation has ideal boundary, then you will obtain an invalid 4-manifold triangulation as a result.

Parameters

base	the 3-manifold triangulation M, as described above.

Returns: the product M x I.

◆ k3()

static Triangulation< 4 > regina::Example< 4 >::k3 ( )

static

Returns a triangulation of the standard K3 surface.

Be warned: this triangulation is fairly large.

Returns: the K3 surface.

◆ rp4()

static Triangulation< 4 > regina::Example< 4 >::rp4 ( )

static

Returns a four-pentachoron triangulation of real projective 4-space.

Returns: real projective 4-space.

◆ s1Bundle()

static Triangulation< 4 > regina::Example< 4 >::s1Bundle ( const Triangulation< 3 > & base )

static

Returns a triangulation of the product M x S1, where M is the given 3-manifold triangulation.

This simply calls iBundle() and then glues together the two copies of M on the boundary.

The product will contain 82 pentachora for each original tetrahedron of M, and will contain many internal vertices. It is highly recommended that you call Triangulation<4>::intelligentSimplify() afterwards if you do not need to preserve the combinatorial structure.

Warning: If the given 3-manifold triangulation has ideal boundary, then you will obtain an invalid 4-manifold triangulation as a result.

Parameters

base	the 3-manifold triangulation M, as described above.

Returns: the product M x S1.

◆ s2xs2()

static Triangulation< 4 > regina::Example< 4 >::s2xs2 ( )

static

Returns a six-pentachoron triangulation of the standard product S² x S².

This triangulation is minimal.

Returns: the standard product of two 2-spheres.

◆ s2xs2Twisted()

static Triangulation< 4 > regina::Example< 4 >::s2xs2Twisted ( )

static

Returns a six-pentachoron triangulation of the twisted product S² x~ S².

This manifold is diffeomorphic to CP² # -CP², where -CP² denotes CP² with its orientation reversed. This triangulation is minimal.

Returns: the twisted product of two 2-spheres.

◆ s3xs1()

Triangulation< 4 > regina::Example< 4 >::s3xs1 ( )

inlinestatic

Returns a two-pentachoron triangulation of the product space S³ x S¹.

This is identical to calling the generic routine sphereBundle().

Returns: the product S³ x S¹.

◆ s3xs1Twisted()

Triangulation< 4 > regina::Example< 4 >::s3xs1Twisted ( )

inlinestatic

Returns a two-pentachoron triangulation of the twisted product space S³ x~ S¹.

This is identical to calling the generic routine twistedSphereBundle().

Returns: the twisted product S³ x~ S¹.

◆ simplicialFourSphere()

Triangulation< 4 > regina::Example< 4 >::simplicialFourSphere ( )

inlinestatic

Returns the standard six-pentachoron triangulation of the 4-sphere as the boundary of a 5-simplex.

This is identical to calling the generic routine simplicialSphere().

Returns: the standard simplicial 4-sphere.

◆ simplicialSphere()

Triangulation< dim > regina::detail::ExampleBase< dim >::simplicialSphere

staticinherited

Returns the standard (dim+2)-simplex triangulation of the dim-sphere as the boundary of a (dim+1)-simplex.

Returns: the standard simplicial dim-sphere.

◆ singleCone()

Triangulation< dim > regina::detail::ExampleFromLowDim< dim, available >::singleCone ( const Triangulation< dim-1 > & base )

staticinherited

Returns a single cone over the given (dim-1)-dimensional triangulation.

If the given triangulation represents the manifold M, then this returns a triangulation of the product M x I that has one real boundary component and one ideal boundary component. The triangulation of the real boundary component will be identical to the original (dim-1)-dimensional triangulation base.

Warning: If the given (dim-1)-dimensional triangulation has any boundary whatsoever (either real or ideal), then unless it is a (dim-1)-ball, you will obtain an invalid dim-manifold triangulation as a result.

Returns: a single cone over the given triangulation.

◆ sphere()

Triangulation< dim > regina::detail::ExampleBase< dim >::sphere

staticinherited

Closed Triangulations.

Returns a two-simplex triangulation of the dim-sphere.

Returns: a two-simplex dim-sphere.

◆ sphereBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::sphereBundle

staticinherited

Returns a two-simplex triangulation of the product space S^(dim-1) x S¹.

Returns: the product S^(dim-1) x S¹.

◆ twistedBallBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::twistedBallBundle

staticinherited

Returns a triangulation of the twisted product space B^(dim-1) x~ S¹.

This will use one simplex in even dimensions, or two simplices in odd dimensions.

Returns: the twisted product B^(dim-1) x~ S¹.

◆ twistedSphereBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::twistedSphereBundle

staticinherited

Returns a two-simplex triangulation of the twisted product space S^(dim-1) x~ S¹.

Returns: the twisted product S^(dim-1) x~ S¹.

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

triangulation/example4.h

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ ball()

◆ ballBundle()

◆ bundleWithMonodromy()

◆ cappellShaneson()

◆ cp2()

◆ doubleCone()

◆ fourSphere()

◆ iBundle()

◆ k3()

◆ rp4()

◆ s1Bundle()

◆ s2xs2()

◆ s2xs2Twisted()

◆ s3xs1()

◆ s3xs1Twisted()

◆ simplicialFourSphere()

◆ simplicialSphere()

◆ singleCone()

◆ sphere()

◆ sphereBundle()

◆ twistedBallBundle()

◆ twistedSphereBundle()