In mathematics, specifically in

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

of manifolds, a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

codimension-one

submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...

F

of a manifold

M

is said to be 2-sided in

M

when there is an

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...

h\colon F\times 1,1 to M

with

h(x,0)=x

for each

x\in F

and ::

h(F\times 1,1 \cap \partial M=h(\partial F\times 1,1

. In other words, if its

normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian m ...

is trivial. This means, for example that a curve in a surface is 2-sided if it has a

tubular neighborhood In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the ...

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 In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...

(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 In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...

has been done – but need not, such as cutting along a curve on the

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

. 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.

References

{{DEFAULTSORT:2-Sided Geometric topology