WordNet

affecting or undertaken by two parties; "a bilateral agreement between the United States and Japan"
having identical parts on each side of an axis (同)isobilateral, bilaterally symmetrical, bilaterally symmetric
having two sides or parts (同)two-sided
with the involvement of two parties or governments; "they worked out an agreement bilaterally"
so as to involve two sides or parts
the property of being symmetrical about a vertical plane (同)bilateralism, bilateral symmetry
take sides for or against; "Who are you widing with?"; "I"m siding against the current candidate"
an extended outer surface of an object; "he turned the box over to examine the bottom side"; "they painted all four sides of the house"
a surface forming part of the outside of an object; "he examined all sides of the crystal"; "dew dripped from the face of the leaf" (同)face
an opinion that is held in opposition to another in an argument or dispute; "there are two sides to every question" (同)position
a lengthwise dressed half of an animals carcass used for food (同)side of meat
a family line of descent; "he gets his brains from his fathers side"
a line segment forming part of the perimeter of a plane figure; "the hypotenuse of a right triangle is always the longest side"
a place within a region identified relative to a center or reference location; "they always sat on the right side of the church"; "he never left my side"
an aspect of something (as contrasted with some other implied aspect); "he was on the heavy side"; "he is on the purchasing side of the business"; "it brought out his better side"
either the left or right half of a body; "he had a pain in his side"
one of two or more contesting groups; "the Confederate side was prepared to attack"
located on a side; "side fences"; "the side porch"
being one more than one; "he received two messages" (同)2, ii
the cardinal number that is the sum of one and one or a numeral representing this number (同)2, II, deuce

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両側のある,左右同形の / 双方の,二派の / (おもに法律で)相互に義務を負う,双務的な
(物体)『面』 / (物体の)『側面』,わき / (紙・レコードなど平らな物の)『面』 / (中心線・分離線から見た)『片側』 / (中心部分から見て)…『側』 / (人間・動物の体の右または左の)『側面』,(特に)横腹,わき腹 / 《単随形で》(人の)『そば』,わき / (物の)端,へり;(図形の)辺 / (競技・争いなどの一方の)『側』,味方 / (事柄の持っている)一面,局面 / (父方または母方の)血統,…方(かた) / 側部の,側面の,わきの / 側部(側面)からの(への) / 二次的な,従の / 〈建物など〉‘に'側面を付ける / (…に賛成の(反対の))側につく《+『with(against)』+『名』》
〈C〉(数の)『2』,2の記号 / 〈U〉2時,2分;2歳 / 《複数扱い》二人;2個 / 〈C〉二つ(二人,2個)1組のもの / 〈C〉(カードゲーム)2の札,(さいころの)2の目 / 2の;二人の;二つの,2個の / 《補語にのみ用いて》2歳の(で)

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/09/25 07:04:52」(JST)

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In topology, a compact codimension one submanifold $F$ of a manifold $M$ is said to be 2-sided in $M$ when there is an embedding

with $h(x,0)=x$ for each $x\in F$ and

.

In other words, if its normal bundle is trivial.

This means, for example that a curve in a surface is 2-sided if it has a tubular neighborhood which is a cartesian product of the curve times an interval.

A submanifold which is not 2-sided is called 1-sided.

Examples

Surfaces

For curves on surfaces, a curve is 2-sided if and only if it preserves orientation, and 1-sided if and only if it reverses orientation: a tubular neighborhood is then a Möbius strip. This can be determined from the class of the curve in the fundamental group of the surface and the orientation character on the fundamental group, which identifies which curves reverse orientation.

An embedded circle in the plane is 2-sided.
An embedded circle generating the fundamental group of the real projective plane (such as an "equator" of the projective plane – the image of an equator for the sphere) is 1-sided, as it is orientation-reversing.

Properties

Cutting along a 2-sided manifold can separate a manifold into two pieces – such as cutting along the equator of a sphere or around the sphere on which a connected sum has been done – but need not, such as cutting along a curve on the torus.

Cutting along a (connected) 1-sided manifold does not separate a manifold, as a point that is locally on one side of the manifold can be connected to a point that is locally on the other side (i.e., just across the submanifold) by passing along an orientation-reversing path.

Cutting along a 1-sided manifold may make a non-orientable manifold orientable – such as cutting along an equator of the real projective plane – but may not, such as cutting along a 1-sided curve in a higher genus non-orientable surface, maybe the simplest example of this is seen when one cut a mobius band along its core curve.

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