Simplicial volume
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== Definition and history == | == Definition and history == | ||
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Revision as of 14:39, 23 March 2010
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 09:51, 1 April 2011 and the changes since publication. |
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1 Definition and history
The simplicial volume is a homotopy invariant of oriented closed connected manifolds that was introduced by Gromov in his proof of Mostow rigidity [Munkholm1980][Gromov1982]. Intuitively, the simplicial volume measures how difficult it is to describe the manifold in question in terms of simplices (with real coefficients):
Definition 1.1. Let be an oriented closed connected manifold of dimension . Then the simplicial volume (also called Gromov norm) of is defined as
Here, denotes the singular chain complex of with real coefficients, and denotes the -norm on the singular chain complex induced from the (unordered) basis given by all singular simplices; i.e., for a chain (in reduced form), the -norm of is given by
2 References
- [Gromov1982] M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982), no.56, 5–99 (1983). MR686042 (84h:53053) Zbl 0516.53046
- [Munkholm1980] H. J. Munkholm, Simplices of maximal volume in hyperbolic space, Gromov's norm, and Gromov's proof of Mostow's rigidity theorem (following Thurston), 788 (1980), 109–124. MR585656 (81k:53046) Zbl 0434.57017
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