Simplicial volume
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manifold in question in terms of simplices (with real coefficients): | manifold in question in terms of simplices (with real coefficients): | ||
− | {{Beginthm|Definition}} | + | {{Beginthm|Definition (Simplicial volume)}} |
Let $M$ be an oriented closed connected manifold of dimension $n$. | Let $M$ be an oriented closed connected manifold of dimension $n$. | ||
Then the '''simplicial volume''' (also called '''Gromov norm''') of $M | Then the '''simplicial volume''' (also called '''Gromov norm''') of $M | ||
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* Moreover, $\|\cdot\|_1$ denotes the '''$\ell^1$-semi-norm''' on singular homology $H_*(\,\cdot\,;\mathbb{R})$ with real coefficients, which is induced by $|\cdot|_1$. More explicitly, if $X$ is a topological space and $\alpha \in H_*(X;\mathbb{R})$, then | * Moreover, $\|\cdot\|_1$ denotes the '''$\ell^1$-semi-norm''' on singular homology $H_*(\,\cdot\,;\mathbb{R})$ with real coefficients, which is induced by $|\cdot|_1$. More explicitly, if $X$ is a topological space and $\alpha \in H_*(X;\mathbb{R})$, then | ||
$$ \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$$ | $$ \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$$ | ||
+ | {{Endthm}} | ||
+ | |||
+ | {{Beginthm|Convention}} | ||
+ | In the following, if not explicitly stated otherwise, all manifolds are topological manifolds and are of non-zero dimension. | ||
{{Endthm}} | {{Endthm}} | ||
Revision as of 17:16, 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 (Simplicial volume) 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
Convention 1.2. In the following, if not explicitly stated otherwise, all manifolds are topological manifolds and are of non-zero dimension.
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.
Hence, all oriented closed connected manifolds admitting a self-map of non-trivial degree (i.e., not equal to , , or ) have vanishing simplicial volume; for instance, the simplicial volume of all
- spheres
- tori
- (odd-dimensional) real projective spaces
- complex projective spaces
is zero.
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
Convention 1.2. In the following, if not explicitly stated otherwise, all manifolds are topological manifolds and are of non-zero dimension.
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.
Hence, all oriented closed connected manifolds admitting a self-map of non-trivial degree (i.e., not equal to , , or ) have vanishing simplicial volume; for instance, the simplicial volume of all
- spheres
- tori
- (odd-dimensional) real projective spaces
- complex projective spaces
is zero.
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
Convention 1.2. In the following, if not explicitly stated otherwise, all manifolds are topological manifolds and are of non-zero dimension.
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.
Hence, all oriented closed connected manifolds admitting a self-map of non-trivial degree (i.e., not equal to , , or ) have vanishing simplicial volume; for instance, the simplicial volume of all
- spheres
- tori
- (odd-dimensional) real projective spaces
- complex projective spaces
is zero.
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