In mathematics , a fundamental class is a producer of the highest homology group of a manifold . In the case of triangulated manifolds, the fundamental class can be represented by the formal sum of the coherently oriented simplices of the triangulation.
Cycles that represent the fundamental class (i.e., whose homology class is the fundamental class) are called fundamental cycles .
Definitions
Closed, orientable manifolds
Let it be a closed, orientable -dimensional manifold. Then
and one calls one of the two producers the fundamental class .
Manifolds with a margin
Let it be a compact , orientable -dimensional manifold with a boundary . Then
and one calls one of the two producers the relative fundamental class .
Non-orientable manifolds
Let it be a closed, not necessarily orientable,
-dimensional manifold. Then
and the producer (i.e., the nontrivial element) is called the -fundamental class.
Local orientations
Let it be a -dimensional manifold. Then applies
for each point . If is closed and orientable, then is
an isomorphism and the image of the fundamental class is called local orientation in .
Non-compact manifolds
Let it be an orientable -dimensional manifold. Then there is a homology class for every compact subset
so that every inclusion of compact subsets maps the class to .
Kronecker mating
The canonical Kronecker pairing between homology and cohomology can be interpreted as follows in the case of -dimensional, closed, orientable manifolds. Let the cohomology class in De Rham cohomology be represented by the differential form , then is
-
.
literature
MJ Greenberg, JR Harper: Algebraic topology , Benjamin / Cummings Publishing Co. Inc. Advanced Book Program, 1981
Web links
Fundamental class (Encyclopedia of Mathematics)