Simplicial volume explained

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]

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Notes and References

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  3. Gabai . David . January 1983 . Foliations and the topology of 3-manifolds . Journal of Differential Geometry . 18 . 3 . 445–503 . 10.4310/jdg/1214437784 . 0022-040X . free.
  4. , pp. 196ff.