A simplicial volume is a topological invariant defined for closed manifolds . First considered by Gromov . Simplicial manifold volume usually indicated .
Let be Is a closed manifold, then
Where - rational coefficients in the representation of its fundamental class in terms of the sum of singular simplexes.
- Gromov's theorem: The simplicial volume of a manifold of constant negative curvature is equal to the ratio of its volume to the volume of a regular infinite simplex in a Lobachevsky space of the same curvature.
- For any varieties and same dimension
- Where denotes a connected amount .
- There are positive numbers and such that if the sum of dimensions then
- Where denotes a direct work .
- For any display
- where indicates the degree of display . In particular:
- If the variety allows display degrees of then .
- For anyone simplicial volume -dimensional sphere equals .