Fundamental class

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 ${\displaystyle M}$ ${\displaystyle n}$

${\displaystyle H_{n}(M;\mathbb {Z} )\cong \mathbb {Z} }$

and one calls one of the two producers the fundamental class . ${\displaystyle \left[M\right]}$

Manifolds with a margin

Let it be a compact , orientable -dimensional manifold with a boundary . Then ${\displaystyle M}$ ${\displaystyle n}$

${\displaystyle H_{n}(M,\partial M;\mathbb {Z} )\cong \mathbb {Z} }$

and one calls one of the two producers the relative fundamental class . ${\displaystyle \left[M,\partial M\right]}$

Non-orientable manifolds

Let it be a closed, not necessarily orientable, -dimensional manifold. Then ${\displaystyle M}$ ${\displaystyle n}$

${\displaystyle H_{n}(M;\mathbb {Z} /2\mathbb {Z} )\cong \mathbb {Z} /2\mathbb {Z} }$

and the producer (i.e., the nontrivial element) is called the -fundamental class. ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$

Local orientations

Let it be a -dimensional manifold. Then applies ${\displaystyle M}$ ${\displaystyle n}$

${\displaystyle H_{n}(M,M\setminus \left\{x\right\};\mathbb {Z} )\cong \mathbb {Z} }$

for each point . If is closed and orientable, then is ${\displaystyle x\in M}$ ${\displaystyle M}$

${\displaystyle i_{*}\colon H_{n}(M;\mathbb {Z} )\to H_{n}(M,M\setminus \left\{x\right\};\mathbb {Z} )}$

an isomorphism and the image of the fundamental class is called local orientation in . ${\displaystyle \left[M\right]}$ ${\displaystyle i_{*}}$ ${\displaystyle x}$

Non-compact manifolds

Let it be an orientable -dimensional manifold. Then there is a homology class for every compact subset ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle K\subset M}$

${\displaystyle \left[M\right]_{K}\in H_{n}(M,M\setminus K;\mathbb {Z} )\cong \mathbb {Z} }$

so that every inclusion of compact subsets maps the class to . ${\displaystyle K_{1}\to K_{2}}$ ${\displaystyle \left[M\right]_{K_{2}}}$ ${\displaystyle \left[M\right]_{K_{1}}}$

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 ${\displaystyle n}$ ${\displaystyle \beta \in H^{n}(M;\mathbb {R} )}$ ${\displaystyle \omega }$

${\displaystyle \langle \beta ,\left[M\right]\rangle =\int _{M}\omega }$.

literature

MJ Greenberg, JR Harper: Algebraic topology , Benjamin / Cummings Publishing Co. Inc. Advanced Book Program, 1981

Web links

Fundamental class (Encyclopedia of Mathematics)