In mathematics, specifically in

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...

-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 S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...

F

of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

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 (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...

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 ...

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 p ...

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 In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...

. This can be determined from the class of the curve in the

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

of the surface and the

orientation character In algebraic topology, a branch of mathematics, an orientation character on a group \pi is a group homomorphism where: :\omega\colon \pi \to \left\ This notion is of particular significance in surgery theory. Motivation Given a manifold ''M'', o ...

on the fundamental group, which identifies which curves reverse orientation. * An embedded circle in the plane is 2-sided. * An embedded circle generating the

of the

real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

(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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

. 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