manu/mldr-zoomed-100char-total-queries-documents

chunk_id stringlengths 3 9	chunk stringlengths 1 100
64_36	Most surfaces encountered in the physical world are orientable. Spheres, planes, and tori are
64_37	orientable, for example. But Möbius strips, real projective planes, and Klein bottles are
64_38	non-orientable. They, as visualized in 3-dimensions, all have just one side. The real projective
64_39	plane and Klein bottle cannot be embedded in R3, only immersed with nice intersections.
64_40	Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a
64_41	one-sided surface would think there is an "other side". The essence of one-sidedness is that the
64_42	ant can crawl from one side of the surface to the "other" without going through the surface or
64_43	flipping over an edge, but simply by crawling far enough.
64_44	In general, the property of being orientable is not equivalent to being two-sided; however, this
64_45	holds when the ambient space (such as R3 above) is orientable. For example, a torus embedded in
64_46	can be one-sided, and a Klein bottle in the same space can be two-sided; here refers to the Klein
64_47	bottle.
64_48	Orientation by triangulation
64_49	Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle
64_50	is glued to at most one other edge. Each triangle is oriented by choosing a direction around the
64_51	perimeter of the triangle, associating a direction to each edge of the triangle. If this is done
64_52	in such a way that, when glued together, neighboring edges are pointing in the opposite direction,
64_53	then this determines an orientation of the surface. Such a choice is only possible if the surface
64_54	is orientable, and in this case there are exactly two different orientations.
64_55	If the figure can be consistently positioned at all points of the surface without turning into its
64_56	mirror image, then this will induce an orientation in the above sense on each of the triangles of
64_57	the triangulation by selecting the direction of each of the triangles based on the order
64_58	red-green-blue of colors of any of the figures in the interior of the triangle.
64_59	This approach generalizes to any n-manifold having a triangulation. However, some 4-manifolds do
64_60	not have a triangulation, and in general for n > 4 some n-manifolds have triangulations that are
64_61	inequivalent.
64_62	Orientability and homology
64_63	If H1(S) denotes the first homology group of a surface S, then S is orientable if and only if H1(S)
64_64	has a trivial torsion subgroup. More precisely, if S is orientable then H1(S) is a free abelian
64_65	group, and if not then H1(S) = F + Z/2Z where F is free abelian, and the Z/2Z factor is generated
64_66	by the middle curve in a Möbius band embedded in S.
64_67	Orientability of manifolds
64_68	Let M be a connected topological n-manifold. There are several possible definitions of what it
64_69	means for M to be orientable. Some of these definitions require that M has extra structure, like
64_70	being differentiable. Occasionally, must be made into a special case. When more than one of
64_71	these definitions applies to M, then M is orientable under one definition if and only if it is
64_72	orientable under the others.
64_73	Orientability of differentiable manifolds
64_74	The most intuitive definitions require that M be a differentiable manifold. This means that the
64_75	transition functions in the atlas of M are C1-functions. Such a function admits a Jacobian
64_76	determinant. When the Jacobian determinant is positive, the transition function is said to be
64_77	orientation preserving. An oriented atlas on M is an atlas for which all transition functions are
64_78	orientation preserving. M is orientable if it admits an oriented atlas. When , an orientation of
64_79	M is a maximal oriented atlas. (When , an orientation of M is a function .)
64_80	Orientability and orientations can also be expressed in terms of the tangent bundle. The tangent
64_81	bundle is a vector bundle, so it is a fiber bundle with structure group . That is, the transition
64_82	functions of the manifold induce transition functions on the tangent bundle which are fiberwise
64_83	linear transformations. If the structure group can be reduced to the group of positive
64_84	determinant matrices, or equivalently if there exists an atlas whose transition functions determine
64_85	an orientation preserving linear transformation on each tangent space, then the manifold M is
64_86	orientable. Conversely, M is orientable if and only if the structure group of the tangent bundle
64_87	can be reduced in this way. Similar observations can be made for the frame bundle.
64_88	Another way to define orientations on a differentiable manifold is through volume forms. A volume
64_89	form is a nowhere vanishing section ω of , the top exterior power of the cotangent bundle of M.
64_90	For example, Rn has a standard volume form given by . Given a volume form on M, the collection of
64_91	all charts for which the standard volume form pulls back to a positive multiple of ω is an
64_92	oriented atlas. The existence of a volume form is therefore equivalent to orientability of the
64_93	manifold.
64_94	Volume forms and tangent vectors can be combined to give yet another description of orientability.
64_95	If is a basis of tangent vectors at a point p, then the basis is said to be right-handed if . A
64_96	transition function is orientation preserving if and only if it sends right-handed bases to
64_97	right-handed bases. The existence of a volume form implies a reduction of the structure group of
64_98	the tangent bundle or the frame bundle to . As before, this implies the orientability of M.
64_99	Conversely, if M is orientable, then local volume forms can be patched together to create a global
64_100	volume form, orientability being necessary to ensure that the global form is nowhere vanishing.
64_101	Homology and the orientability of general manifolds
64_102	At the heart of all the above definitions of orientability of a differentiable manifold is the
64_103	notion of an orientation preserving transition function. This raises the question of what exactly
64_104	such transition functions are preserving. They cannot be preserving an orientation of the manifold
64_105	because an orientation of the manifold is an atlas, and it makes no sense to say that a transition
64_106	function preserves or does not preserve an atlas of which it is a member.
64_107	This question can be resolved by defining local orientations. On a one-dimensional manifold, a
64_108	local orientation around a point p corresponds to a choice of left and right near that point. On a
64_109	two-dimensional manifold, it corresponds to a choice of clockwise and counter-clockwise. These two
64_110	situations share the common feature that they are described in terms of top-dimensional behavior
64_111	near p but not at p. For the general case, let M be a topological n-manifold. A local orientation
64_112	of M around a point p is a choice of generator of the group
64_113	To see the geometric significance of this group, choose a chart around p. In that chart there is a
64_114	neighborhood of p which is an open ball B around the origin O. By the excision theorem, is
64_115	isomorphic to . The ball B is contractible, so its homology groups vanish except in degree zero,
64_116	and the space is an -sphere, so its homology groups vanish except in degrees and . A computation
64_117	with the long exact sequence in relative homology shows that the above homology group is isomorphic
64_118	to . A choice of generator therefore corresponds to a decision of whether, in the given chart, a
64_119	sphere around p is positive or negative. A reflection of through the origin acts by negation on ,
64_120	so the geometric significance of the choice of generator is that it distinguishes charts from their
64_121	reflections.
64_122	On a topological manifold, a transition function is orientation preserving if, at each point p in
64_123	its domain, it fixes the generators of . From here, the relevant definitions are the same as in
64_124	the differentiable case. An oriented atlas is one for which all transition functions are
64_125	orientation preserving, M is orientable if it admits an oriented atlas, and when , an orientation
64_126	of M is a maximal oriented atlas.
64_127	Intuitively, an orientation of M ought to define a unique local orientation of M at each point.
64_128	This is made precise by noting that any chart in the oriented atlas around p can be used to
64_129	determine a sphere around p, and this sphere determines a generator of . Moreover, any other chart
64_130	around p is related to the first chart by an orientation preserving transition function, and this
64_131	implies that the two charts yield the same generator, whence the generator is unique.
64_132	Purely homological definitions are also possible. Assuming that M is closed and connected, M is
64_133	orientable if and only if the nth homology group is isomorphic to the integers Z. An orientation
64_134	of M is a choice of generator of this group. This generator determines an oriented atlas by
64_135	fixing a generator of the infinite cyclic group and taking the oriented charts to be those for

64_36

Most surfaces encountered in the physical world are orientable. Spheres, planes, and tori are

64_37

orientable, for example. But Möbius strips, real projective planes, and Klein bottles are

64_38

non-orientable. They, as visualized in 3-dimensions, all have just one side. The real projective

64_39

plane and Klein bottle cannot be embedded in R3, only immersed with nice intersections.

64_40

Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a

64_41

one-sided surface would think there is an "other side". The essence of one-sidedness is that the

64_42

ant can crawl from one side of the surface to the "other" without going through the surface or

64_43

flipping over an edge, but simply by crawling far enough.

64_44

In general, the property of being orientable is not equivalent to being two-sided; however, this

64_45

holds when the ambient space (such as R3 above) is orientable. For example, a torus embedded in

64_46

can be one-sided, and a Klein bottle in the same space can be two-sided; here refers to the Klein

64_47

bottle.

64_48

Orientation by triangulation

64_49

Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle

64_50

is glued to at most one other edge. Each triangle is oriented by choosing a direction around the

64_51

perimeter of the triangle, associating a direction to each edge of the triangle. If this is done

64_52

in such a way that, when glued together, neighboring edges are pointing in the opposite direction,

64_53

then this determines an orientation of the surface. Such a choice is only possible if the surface

64_54

is orientable, and in this case there are exactly two different orientations.

64_55

If the figure can be consistently positioned at all points of the surface without turning into its

64_56

mirror image, then this will induce an orientation in the above sense on each of the triangles of

64_57

the triangulation by selecting the direction of each of the triangles based on the order

64_58

red-green-blue of colors of any of the figures in the interior of the triangle.

64_59

This approach generalizes to any n-manifold having a triangulation. However, some 4-manifolds do

64_60

not have a triangulation, and in general for n > 4 some n-manifolds have triangulations that are

64_61

inequivalent.

64_62

Orientability and homology

64_63

If H1(S) denotes the first homology group of a surface S, then S is orientable if and only if H1(S)

64_64

has a trivial torsion subgroup. More precisely, if S is orientable then H1(S) is a free abelian

64_65

group, and if not then H1(S) = F + Z/2Z where F is free abelian, and the Z/2Z factor is generated

64_66

by the middle curve in a Möbius band embedded in S.

64_67

Orientability of manifolds

64_68

Let M be a connected topological n-manifold. There are several possible definitions of what it

64_69

means for M to be orientable. Some of these definitions require that M has extra structure, like

64_70

being differentiable. Occasionally, must be made into a special case. When more than one of

64_71

these definitions applies to M, then M is orientable under one definition if and only if it is

64_72

orientable under the others.

64_73

Orientability of differentiable manifolds

64_74

The most intuitive definitions require that M be a differentiable manifold. This means that the

64_75

transition functions in the atlas of M are C1-functions. Such a function admits a Jacobian

64_76

determinant. When the Jacobian determinant is positive, the transition function is said to be

64_77

orientation preserving. An oriented atlas on M is an atlas for which all transition functions are

64_78

orientation preserving. M is orientable if it admits an oriented atlas. When , an orientation of

64_79

M is a maximal oriented atlas. (When , an orientation of M is a function .)

64_80

Orientability and orientations can also be expressed in terms of the tangent bundle. The tangent

64_81

bundle is a vector bundle, so it is a fiber bundle with structure group . That is, the transition

64_82

functions of the manifold induce transition functions on the tangent bundle which are fiberwise

64_83

linear transformations. If the structure group can be reduced to the group of positive

64_84

determinant matrices, or equivalently if there exists an atlas whose transition functions determine

64_85

an orientation preserving linear transformation on each tangent space, then the manifold M is

64_86

orientable. Conversely, M is orientable if and only if the structure group of the tangent bundle

64_87

can be reduced in this way. Similar observations can be made for the frame bundle.

64_88

Another way to define orientations on a differentiable manifold is through volume forms. A volume

64_89

form is a nowhere vanishing section ω of , the top exterior power of the cotangent bundle of M.

64_90

For example, Rn has a standard volume form given by . Given a volume form on M, the collection of

64_91

all charts for which the standard volume form pulls back to a positive multiple of ω is an

64_92

oriented atlas. The existence of a volume form is therefore equivalent to orientability of the

64_93

manifold.

64_94

Volume forms and tangent vectors can be combined to give yet another description of orientability.

64_95

If is a basis of tangent vectors at a point p, then the basis is said to be right-handed if . A

64_96

transition function is orientation preserving if and only if it sends right-handed bases to

64_97

right-handed bases. The existence of a volume form implies a reduction of the structure group of

64_98

the tangent bundle or the frame bundle to . As before, this implies the orientability of M.

64_99

Conversely, if M is orientable, then local volume forms can be patched together to create a global

64_100

volume form, orientability being necessary to ensure that the global form is nowhere vanishing.

64_101

Homology and the orientability of general manifolds

64_102

At the heart of all the above definitions of orientability of a differentiable manifold is the

64_103

notion of an orientation preserving transition function. This raises the question of what exactly

64_104

such transition functions are preserving. They cannot be preserving an orientation of the manifold

64_105

because an orientation of the manifold is an atlas, and it makes no sense to say that a transition

64_106

function preserves or does not preserve an atlas of which it is a member.

64_107

This question can be resolved by defining local orientations. On a one-dimensional manifold, a

64_108

local orientation around a point p corresponds to a choice of left and right near that point. On a

64_109

two-dimensional manifold, it corresponds to a choice of clockwise and counter-clockwise. These two

64_110

situations share the common feature that they are described in terms of top-dimensional behavior

64_111

near p but not at p. For the general case, let M be a topological n-manifold. A local orientation

64_112

of M around a point p is a choice of generator of the group

64_113

To see the geometric significance of this group, choose a chart around p. In that chart there is a

64_114

neighborhood of p which is an open ball B around the origin O. By the excision theorem, is

64_115

isomorphic to . The ball B is contractible, so its homology groups vanish except in degree zero,

64_116

and the space is an -sphere, so its homology groups vanish except in degrees and . A computation

64_117

with the long exact sequence in relative homology shows that the above homology group is isomorphic

64_118

to . A choice of generator therefore corresponds to a decision of whether, in the given chart, a

64_119

sphere around p is positive or negative. A reflection of through the origin acts by negation on ,

64_120

so the geometric significance of the choice of generator is that it distinguishes charts from their

64_121

reflections.

64_122

On a topological manifold, a transition function is orientation preserving if, at each point p in

64_123

its domain, it fixes the generators of . From here, the relevant definitions are the same as in

64_124

the differentiable case. An oriented atlas is one for which all transition functions are

64_125

orientation preserving, M is orientable if it admits an oriented atlas, and when , an orientation

64_126

of M is a maximal oriented atlas.

64_127

Intuitively, an orientation of M ought to define a unique local orientation of M at each point.

64_128

This is made precise by noting that any chart in the oriented atlas around p can be used to

64_129

determine a sphere around p, and this sphere determines a generator of . Moreover, any other chart

64_130

around p is related to the first chart by an orientation preserving transition function, and this

64_131

implies that the two charts yield the same generator, whence the generator is unique.

64_132

Purely homological definitions are also possible. Assuming that M is closed and connected, M is

64_133

orientable if and only if the nth homology group is isomorphic to the integers Z. An orientation

64_134

of M is a choice of generator of this group. This generator determines an oriented atlas by

64_135

fixing a generator of the infinite cyclic group and taking the oriented charts to be those for