regina::detail::ExampleFromLowDim< dim, false > Class Template Reference

Helper class that builds various dim-dimensional triangulations from (dim-1)-dimensional triangulations. More...

#include <triangulation/detail/example.h>

Static Public Member Functions
static Triangulation< dim >	doubleCone (const Triangulation< dim-1 > &base)
	Returns a double cone over the given (dim-1)-dimensional triangulation.
static Triangulation< dim >	singleCone (const Triangulation< dim-1 > &base)
	Returns a single cone over the given (dim-1)-dimensional triangulation.

Detailed Description

template<int dim>
class regina::detail::ExampleFromLowDim< dim, false >

Helper class that builds various dim-dimensional triangulations from (dim-1)-dimensional triangulations.

See the general ExampleFromLowDim template notes for further details.

This specialisation is used for dimensions in which (dim-1)-dimensional triangulations are not supported. As a result, this specialised class is empty.

Member Function Documentation

doubleCone()

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

static

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 × 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 × I we "remove" the vertices at the apex of each cone.

If the given 3-dimensional triangulation is oriented, then the resulting 4-dimensional triangulation will be oriented also.

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.

singleCone()

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

static

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 × 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.

If the given 3-dimensional triangulation is oriented, then the resulting 4-dimensional triangulation will be oriented also.

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.

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

• triangulation/detail/example.h