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

  1. ^ a b Benedetti, Riccardo; Petronio, Carlo (1992), Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, p. 105, doi:10.1007/978-3-642-58158-8, ISBN 3-540-55534-X, MR 1219310.
  2. ^ Ratcliffe, John G. (2006), Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149 (2nd ed.), Berlin: Springer, p. 555, doi:10.1007/978-1-4757-4013-4, ISBN 978-0387-33197-3, MR 2249478, S2CID 123040867.
  3. ^ Gabai, David (January 1983). "Foliations and the topology of 3-manifolds". Journal of Differential Geometry. 18 (3): 445–503. doi:10.4310/jdg/1214437784. ISSN 0022-040X.
  4. ^ Benedetti & Petronio (1992), pp. 196ff.


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