Simplicial volume
In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes.
Given a closed and oriented manifold, one defines the simplicial norm by minimizing the sum of the absolute values of the coefficients over all singular chains homologous to a given cycle. The simplicial volume is the simplicial norm of the fundamental class.[1][2]
It is named after Mikhail Gromov, who introduced it in 1982. With William Thurston, he proved that the simplicial volume of a finite volume hyperbolic manifold is proportional to the hyperbolic volume.[1]
The simplicial volume is equal to twice the Thurston norm.[3]
Thurston also used the simplicial volume to prove that hyperbolic volume decreases under hyperbolic Dehn surgery.[4]
References
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- Michael Gromov. Volume and bounded cohomology. Template:Free access Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.
External links
- Simplicial volume at the Manifold Atlas.