# Simplicial volume

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In the mathematical field of [geometric topology](/source/Geometric_topology), the **simplicial volume** (also called **Gromov norm**) is a measure of the topological complexity of a [manifold](/source/Manifold). More generally, the **simplicial norm** measures the complexity of [homology classes](/source/Homology_class).

Given a [closed](/source/Closed_manifold) 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](/source/Fundamental_class).[1][2]

It is named after [Mikhail Gromov](/source/Mikhail_Gromov_(mathematician)), who introduced it in 1982. With [William Thurston](/source/William_Thurston), he proved that the simplicial volume of a finite volume [hyperbolic manifold](/source/Hyperbolic_manifold) is proportional to the [hyperbolic volume](/source/Hyperbolic_volume).[1]

The simplicial volume is equal to twice the [Thurston norm](/source/Thurston_norm).[3]

Thurston also used the simplicial volume to prove that hyperbolic volume decreases under [hyperbolic Dehn surgery](/source/Hyperbolic_Dehn_surgery).[4]

## References

1. Benedetti, Riccardo & Petronio, Carlo (1992), [*Lectures on hyperbolic geometry*](https://books.google.com/books?id=iTbmytIqdpcC&pg=PA105), Universitext, Springer-Verlag, Berlin, p. 105, [doi:10.1007/978-3-642-58158-8](https://doi.org/10.1007/978-3-642-58158-8). ISBN 3-540-55534-X. MR 1219310.

1. Ratcliffe, John G. (2006), [*Foundations of hyperbolic manifolds*](https://books.google.com/books?id=JV9m8o-ok6YC&pg=PA555), Vol. 149, Graduate Texts in Mathematics, 2nd ed., Berlin: Springer, p. 555, [doi:10.1007/978-1-4757-4013-4](https://doi.org/10.1007/978-1-4757-4013-4). ISBN 978-0387-33197-3. MR 2249478. [S2CID 123040867](https://api.semanticscholar.org/CorpusID:123040867).

1. Gabai, David (January 1983). "Foliations and the topology of 3-manifolds". *Journal of Differential Geometry*. **18** (3): 445–503. [doi:10.4310/jdg/1214437784](https://doi.org/10.4310/jdg/1214437784). [ISSN 0022-040X](https://www.worldcat.org/issn/0022-040X)

1. Benedetti & Petronio (1992), pp. 196ff.

- Michael Gromov. [*Volume and bounded cohomology.*](http://www.numdam.org/item/PMIHES_1982__56__5_0/) Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.

## External links

- [Simplicial volume](http://www.map.mpim-bonn.mpg.de/Simplicial_volume) at the Manifold Atlas.

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