Simplicial volume
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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
where is the fundamental class of with real coefficients.
- Here, denotes the -norm on the singular chain complex with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space and a chain (in reduced form), the -norm of is given by
- Moreover, denotes the -semi-norm on singular homology with real coefficients, which is induced by . More explicitly, if is a topological space and , then
2 Functoriality and elementary examples
The -semi-norm is functorial in the following sense [Gromov1999]:
Proposition 2.1. If is a continuous map of topological spaces and , then
as can be seen by inspecting the definition of and of .
Corollary 2.2.
- Let be a map of oriented closed connected manifolds of the same dimension. Then
- Because homotopy equivalences of oriented closed connected manifolds have degree or , it follows that the simplicial volume indeed is a homotopy invariant of oriented closed connected manifolds.
3 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
- [Gromov1999] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser Boston Inc., Boston, MA, 1999. MR1699320 (2000d:53065) Zbl 1113.53001
- [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
This page has not been refereed. The information given here might be incomplete or provisional. |
where is the fundamental class of with real coefficients.
- Here, denotes the -norm on the singular chain complex with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space and a chain (in reduced form), the -norm of is given by
- Moreover, denotes the -semi-norm on singular homology with real coefficients, which is induced by . More explicitly, if is a topological space and , then
2 Functoriality and elementary examples
The -semi-norm is functorial in the following sense [Gromov1999]:
Proposition 2.1. If is a continuous map of topological spaces and , then
as can be seen by inspecting the definition of and of .
Corollary 2.2.
- Let be a map of oriented closed connected manifolds of the same dimension. Then
- Because homotopy equivalences of oriented closed connected manifolds have degree or , it follows that the simplicial volume indeed is a homotopy invariant of oriented closed connected manifolds.
3 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
- [Gromov1999] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser Boston Inc., Boston, MA, 1999. MR1699320 (2000d:53065) Zbl 1113.53001
- [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
This page has not been refereed. The information given here might be incomplete or provisional. |