There are many definitions of an orientation of a manifold. The most general, in the sense that it doesn’t require any extra on the manifold, is based on (co-)homology theory. For this article manifold means a connected, topological manifold possibly with boundary.
Let be a closed, –dimensional manifold. Then the top dimensional homology group of , is either trivial () or isomorphic to .
A closed –manifold is called orientable if its top homology group is isomorphic to the integers. An orientation of is a choice of a particular isomorphism
An oriented manifold is a (necessarily orientable) manifold endowed with an orientation. If is an oriented manifold then is called the fundamental class of , or the orientation class of , and is denoted by .
The above definition could be given using cohomology instead of homology.
The top dimensional homology of a non-closed manifold is always trivial, so it is trickier to define orientation for those beasts. One approach (which we will not follow) is to use special kind of homology (for example relative to the boundary for compact manifolds with boundary). The approach we follow defines (global) orientation as compatible fitting together of local orientations. We start with manifolds without boundary.
Let be an -manifold without boundary and . Then the relative homology group
Let be an -manifold and . An orientation of at is a choice of an isomorphism
to make precise the notion of nicely fitting together of orientations at points, is to require that for nearby points the orientations are defined in a way.
Let be an open subset of that is homeomorphic to . A local orientation of on is a choice of an isomorphism
Now notice that with as above and the inclusion
a map (actually isomorphism)
and therefore a local orientation at (by composing with the above isomorphism) an orientation at each point . It is to declare that all these orientations fit nicely together.
Let be a manifold with non-empty boundary, . is called orientable if its double
is orientable, where denotes gluing along the boundary.
An orientation of is determined by an orientation of .
|Date of creation||2013-03-22 12:56:24|
|Last modified on||2013-03-22 12:56:24|
|Last modified by||PrimeFan (13766)|