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 (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
![\displaystyle \|M\| := \bigl\| [M] \bigr\|_1 = \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_n(M;\mathbb{R})$ is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0},](/images/math/c/2/2/c22511eadabaaaa5caa7565812d6977a.png)
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
![\displaystyle |c|_1 := \sum_{j=0}^k |a_j|.](/images/math/c/8/d/c8d474e1f10893ce927aad5b02951cbe.png)
- Moreover,
denotes the
-semi-norm on singular homology
with real coefficients, which is induced by
. More explicitly, if
is a topological space and
, then
![\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.](/images/math/c/1/0/c109b20bb54f3169d86095ba15ddee57.png)
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
![\displaystyle \bigl\| H_*(f;\mathbb{R}) (\alpha) \bigr\|_1 \leq \|\alpha\|_1,](/images/math/8/c/8/8c807b3de755533551b84e276514ae1e.png)
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
![\displaystyle |\deg f| \cdot \|N\| \leq \|M\|.](/images/math/6/c/f/6cf13a91a3507c652ffcbac5d72b7d9d.png)
- 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. |
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle \|M\| := \bigl\| [M] \bigr\|_1 = \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_n(M;\mathbb{R})$ is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0},](/images/math/c/2/2/c22511eadabaaaa5caa7565812d6977a.png)
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
![\displaystyle |c|_1 := \sum_{j=0}^k |a_j|.](/images/math/c/8/d/c8d474e1f10893ce927aad5b02951cbe.png)
- Moreover,
denotes the
-semi-norm on singular homology
with real coefficients, which is induced by
. More explicitly, if
is a topological space and
, then
![\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.](/images/math/c/1/0/c109b20bb54f3169d86095ba15ddee57.png)
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
![\displaystyle \bigl\| H_*(f;\mathbb{R}) (\alpha) \bigr\|_1 \leq \|\alpha\|_1,](/images/math/8/c/8/8c807b3de755533551b84e276514ae1e.png)
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
![\displaystyle |\deg f| \cdot \|N\| \leq \|M\|.](/images/math/6/c/f/6cf13a91a3507c652ffcbac5d72b7d9d.png)
- 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. |
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle \|M\| := \bigl\| [M] \bigr\|_1 = \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_n(M;\mathbb{R})$ is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0},](/images/math/c/2/2/c22511eadabaaaa5caa7565812d6977a.png)
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
![\displaystyle |c|_1 := \sum_{j=0}^k |a_j|.](/images/math/c/8/d/c8d474e1f10893ce927aad5b02951cbe.png)
- Moreover,
denotes the
-semi-norm on singular homology
with real coefficients, which is induced by
. More explicitly, if
is a topological space and
, then
![\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.](/images/math/c/1/0/c109b20bb54f3169d86095ba15ddee57.png)
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
![\displaystyle \bigl\| H_*(f;\mathbb{R}) (\alpha) \bigr\|_1 \leq \|\alpha\|_1,](/images/math/8/c/8/8c807b3de755533551b84e276514ae1e.png)
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
![\displaystyle |\deg f| \cdot \|N\| \leq \|M\|.](/images/math/6/c/f/6cf13a91a3507c652ffcbac5d72b7d9d.png)
- 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. |
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle \|M\| := \bigl\| [M] \bigr\|_1 = \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_n(M;\mathbb{R})$ is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0},](/images/math/c/2/2/c22511eadabaaaa5caa7565812d6977a.png)
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
![\displaystyle |c|_1 := \sum_{j=0}^k |a_j|.](/images/math/c/8/d/c8d474e1f10893ce927aad5b02951cbe.png)
- Moreover,
denotes the
-semi-norm on singular homology
with real coefficients, which is induced by
. More explicitly, if
is a topological space and
, then
![\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.](/images/math/c/1/0/c109b20bb54f3169d86095ba15ddee57.png)