Simplicial volume
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− | + | == "Computing" simplicial volume == | |
− | + | ||
− | + | ||
− | + | ||
− | + | <wikitex> | |
+ | In most cases, trying to compute the simplicial volume by inspecting the definition proves to be futile; | ||
+ | the two main sources for non-trivial estimates of the simplicial volume of given manifolds are: | ||
+ | * ''Geometric'': The connection between simplicial volume and Riemannian geometry [[Simplicial volume#Simplicial volume and Riemannian geometry|see below]]. | ||
+ | * ''Algebraic'': The connection between simplicial volume and bounded cohomology [[Simplicial volume#Simplicial volume and bounded cohomology|see below]]. | ||
=== Simplicial volume and Riemannian geometry === | === Simplicial volume and Riemannian geometry === | ||
+ | |||
+ | A fascinating aspect of the simplicial volume is that it is a ''homotopy'' invariant encoding non-trivial information about the Riemannian volume. The most fundamental result of this type is Gromov's lower bound of the minimal volume in terms of the simplicial volume {{cite|Gromov1982|Section 0.5}}: | ||
+ | |||
+ | {{Beginthm|Theorem (Simplicial volume and minimal volume)}} | ||
+ | For all oriented closed connected smooth $n$-manifolds $M$ we have | ||
+ | $$ \|M\| \leq (n-1)^n \cdot n! \cdot \mathop{\mathrm{minvol}}(M).$$ | ||
+ | {{Endthm}} | ||
+ | |||
+ | The '''minimal volume''' {{cite|Gromov1982}} of a complete smooth manifold $M$ is defined as | ||
+ | $$ \mathop{\mathrm{minvol}}(M) := \inf \bigl\{ \mathop{\mathrm{vol}}(M,g) \bigm| \text{$g$ is a Riemannian metric on~$M$ with~$|\mathop{\mathrm{sec}}(g)| \leq 1$}\bigr\}.$$ | ||
+ | |||
+ | <!-- | ||
+ | add background references for minimal volume | ||
+ | --> | ||
+ | |||
+ | <!-- continue here --> | ||
=== Simplicial volume and bounded cohomology === | === Simplicial volume and bounded cohomology === | ||
+ | |||
+ | A more algebraic approach to the simplicial volume is based on the following observation {{cite|Gromov1982|p. 17}}{{cite|Benedetti&Petronio1992|F.2.2}} (see below for an explanation of the notation): | ||
+ | |||
+ | {{Beginthm|Proposition (Duality principle)}} | ||
+ | Let $X$ be a topological space, let $n \in \mathbb{N}$, and let $\alpha \in H_n(X;\mathbb{R})$. Then | ||
+ | $$\begin{aligned} | ||
+ | \|\alpha\|_1 &= \sup \Bigl\{ \frac{1}{\|\varphi\|_\infty} | ||
+ | \Bigm| \varphi \in H^n(X;\mathbb{R}),~\langle \varphi, \alpha\rangle = 1 | ||
+ | \Bigr\} \\ | ||
+ | &= \sup \Bigl\{ \frac{1}{\|\varphi\|_\infty} | ||
+ | \Bigm| \varphi \in H_b^n(X;\mathbb{R}),~\langle \varphi, \alpha\rangle = 1 | ||
+ | \Bigr\}. | ||
+ | \end{aligned}$$ | ||
+ | {{Endthm}} | ||
+ | |||
+ | {{Beginthm|Corollary}} | ||
+ | Let $M$ be an oriented closed connected $n$-manifold. Then, where $[M]^* \in H^n(M;\mathbb{R})$ denotes the cohomology class dual to the real fundamental class of $M$: | ||
+ | $$\begin{aligned} | ||
+ | \| M \| & = \frac{1}{\bigl\| [M]^* \bigr\|_\infty}\\ | ||
+ | & = \sup \Bigl\{ \frac{1}{\|\varphi\|_\infty} | ||
+ | \Bigm| \varphi \in H_b^n(M;\mathbb{R}),~c_M(\varphi) = [M]^* | ||
+ | \Bigr\}. | ||
+ | \end{aligned} | ||
+ | $$ | ||
+ | {{Endthm}} | ||
+ | |||
+ | For the sake of completeness, we review the definition of ''bounded cohomology'' of topological spaces: Let $X$ be a topological space, and let $n \in \mathbb{N}$. | ||
+ | <ul> | ||
+ | <li> If $f \in C^n(X;\mathbb{R})$ is a cochain, then we write | ||
+ | $$ |f|_\infty := \sup_{\sigma \in \mathop{\mathrm{map}} (\Delta^n,X)} |f(\sigma)| | ||
+ | \in \mathbb{R}_{\geq 0} \cup \{\infty\}. | ||
+ | $$ | ||
+ | If $|f|_\infty < \infty$, then $f$ is a '''bounded cochain'''. | ||
+ | </li> | ||
+ | </ul> | ||
+ | * We write $C_b^n(X;\mathbb{R}) := \bigl\{ f \bigm| f \in C^n(X;\mathbb{R}),~|f|_\infty < \infty$ for the subspace of bounded cochains. Notice that $C_b^*(X;\mathbb{R})$ is a subcomplex of the singular cochain complex, called the '''bounded cochain complex''' of $X$. | ||
+ | * The cohomology $H^*_b(X;\mathbb{R})$ of $C^*(X;\mathbb{R})$ is the '''bounded cohomology''' of $X$. | ||
+ | * The norm $|\cdot|_\infty$ on the bounded cochain complex induces a semi-norm on bounded cohomology: If $\varphi \in H^n_b(X;\mathbb{R})$, then | ||
+ | $$ \|\varphi\|_\infty := \bigl\{ |f|_\infty | ||
+ | \bigm| \text{$f \in C^n_b(X;\mathbb{R})$ is a cocycle representing~$\varphi$} | ||
+ | \bigr\}. | ||
+ | $$ | ||
+ | * The inclusion $C_b^*(X;\mathbb{R}) \hookrightarrow C^*(X;\mathbb{R})$ induces a homomorphism $c_X \colon H^*_b(X;\mathbb{R}) \longrightarrow H^*(X;\mathbb{R})$, the '''comparison map'''. | ||
+ | |||
+ | <!-- add refs --> | ||
+ | Bounded cohomology was originally introduced by Trauber. Gromov further developed bounded cohomology and studied its relation with the (Riemannian) volume of manifolds {{cite|Gromov1982}}. A more algebraic approach to bounded cohomology was subsequently developed by Brooks, Ivanov, Noskov, Monod, and Bühler. | ||
+ | |||
+ | </wikitex> | ||
+ | |||
+ | |||
+ | |||
+ | <!-- COMMENT: | ||
+ | |||
+ | 2do! | ||
+ | add references | ||
+ | complete the following sections | ||
+ | add intro/history | ||
== Further examples and properties == | == Further examples and properties == |
Revision as of 14:30, 7 June 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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Contents |
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 (Functoriality of the -semi-norm) 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 "Computing" simplicial volume
In most cases, trying to compute the simplicial volume by inspecting the definition proves to be futile;
the two main sources for non-trivial estimates of the simplicial volume of given manifolds are:
- Geometric: The connection between simplicial volume and Riemannian geometry see below.
- Algebraic: The connection between simplicial volume and bounded cohomology see below.
1 Simplicial volume and Riemannian geometry
A fascinating aspect of the simplicial volume is that it is a homotopy invariant encoding non-trivial information about the Riemannian volume. The most fundamental result of this type is Gromov's lower bound of the minimal volume in terms of the simplicial volume [Gromov1982, Section 0.5]:
Theorem (Simplicial volume and minimal volume) 3.1. For all oriented closed connected smooth -manifolds we have
The minimal volume [Gromov1982] of a complete smooth manifold is defined as
2 Simplicial volume and bounded cohomology
A more algebraic approach to the simplicial volume is based on the following observation [Gromov1982, p. 17][Benedetti&Petronio1992, F.2.2] (see below for an explanation of the notation):
Proposition (Duality principle) 3.2. Let be a topological space, let , and let . Then
Corollary 3.3. Let be an oriented closed connected -manifold. Then, where denotes the cohomology class dual to the real fundamental class of :
For the sake of completeness, we review the definition of bounded cohomology of topological spaces: Let be a topological space, and let .
- If is a cochain, then we write
If , then is a bounded cochain.
- We write for the subspace of bounded cochains. Notice that is a subcomplex of the singular cochain complex, called the bounded cochain complex of .
- The cohomology of is the bounded cohomology of .
- The norm on the bounded cochain complex induces a semi-norm on bounded cohomology: If , then
- The inclusion induces a homomorphism , the comparison map.
Bounded cohomology was originally introduced by Trauber. Gromov further developed bounded cohomology and studied its relation with the (Riemannian) volume of manifolds [Gromov1982]. A more algebraic approach to bounded cohomology was subsequently developed by Brooks, Ivanov, Noskov, Monod, and Bühler.
4 References
- [Benedetti&Petronio1992] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Springer-Verlag, Berlin, 1992. MR1219310 (94e:57015) Zbl 0768.51018
- [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 (Functoriality of the -semi-norm) 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 "Computing" simplicial volume
In most cases, trying to compute the simplicial volume by inspecting the definition proves to be futile;
the two main sources for non-trivial estimates of the simplicial volume of given manifolds are:
- Geometric: The connection between simplicial volume and Riemannian geometry see below.
- Algebraic: The connection between simplicial volume and bounded cohomology see below.
1 Simplicial volume and Riemannian geometry
A fascinating aspect of the simplicial volume is that it is a homotopy invariant encoding non-trivial information about the Riemannian volume. The most fundamental result of this type is Gromov's lower bound of the minimal volume in terms of the simplicial volume [Gromov1982, Section 0.5]:
Theorem (Simplicial volume and minimal volume) 3.1. For all oriented closed connected smooth -manifolds we have
The minimal volume [Gromov1982] of a complete smooth manifold is defined as
2 Simplicial volume and bounded cohomology
A more algebraic approach to the simplicial volume is based on the following observation [Gromov1982, p. 17][Benedetti&Petronio1992, F.2.2] (see below for an explanation of the notation):
Proposition (Duality principle) 3.2. Let be a topological space, let , and let . Then
Corollary 3.3. Let be an oriented closed connected -manifold. Then, where denotes the cohomology class dual to the real fundamental class of :
For the sake of completeness, we review the definition of bounded cohomology of topological spaces: Let be a topological space, and let .
- If is a cochain, then we write
If , then is a bounded cochain.
- We write for the subspace of bounded cochains. Notice that is a subcomplex of the singular cochain complex, called the bounded cochain complex of .
- The cohomology of is the bounded cohomology of .
- The norm on the bounded cochain complex induces a semi-norm on bounded cohomology: If , then
- The inclusion induces a homomorphism , the comparison map.
Bounded cohomology was originally introduced by Trauber. Gromov further developed bounded cohomology and studied its relation with the (Riemannian) volume of manifolds [Gromov1982]. A more algebraic approach to bounded cohomology was subsequently developed by Brooks, Ivanov, Noskov, Monod, and Bühler.
4 References
- [Benedetti&Petronio1992] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Springer-Verlag, Berlin, 1992. MR1219310 (94e:57015) Zbl 0768.51018
- [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 (Functoriality of the -semi-norm) 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 "Computing" simplicial volume
In most cases, trying to compute the simplicial volume by inspecting the definition proves to be futile;
the two main sources for non-trivial estimates of the simplicial volume of given manifolds are:
- Geometric: The connection between simplicial volume and Riemannian geometry see below.
- Algebraic: The connection between simplicial volume and bounded cohomology see below.
1 Simplicial volume and Riemannian geometry
A fascinating aspect of the simplicial volume is that it is a homotopy invariant encoding non-trivial information about the Riemannian volume. The most fundamental result of this type is Gromov's lower bound of the minimal volume in terms of the simplicial volume [Gromov1982, Section 0.5]:
Theorem (Simplicial volume and minimal volume) 3.1. For all oriented closed connected smooth -manifolds we have
The minimal volume [Gromov1982] of a complete smooth manifold is defined as
2 Simplicial volume and bounded cohomology
A more algebraic approach to the simplicial volume is based on the following observation [Gromov1982, p. 17][Benedetti&Petronio1992, F.2.2] (see below for an explanation of the notation):
Proposition (Duality principle) 3.2. Let be a topological space, let , and let . Then
Corollary 3.3. Let be an oriented closed connected -manifold. Then, where denotes the cohomology class dual to the real fundamental class of :
For the sake of completeness, we review the definition of bounded cohomology of topological spaces: Let be a topological space, and let .
- If is a cochain, then we write
If , then is a bounded cochain.
- We write for the subspace of bounded cochains. Notice that is a subcomplex of the singular cochain complex, called the bounded cochain complex of .
- The cohomology of is the bounded cohomology of .
- The norm on the bounded cochain complex induces a semi-norm on bounded cohomology: If , then
- The inclusion induces a homomorphism , the comparison map.
Bounded cohomology was originally introduced by Trauber. Gromov further developed bounded cohomology and studied its relation with the (Riemannian) volume of manifolds [Gromov1982]. A more algebraic approach to bounded cohomology was subsequently developed by Brooks, Ivanov, Noskov, Monod, and Bühler.
4 References
- [Benedetti&Petronio1992] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Springer-Verlag, Berlin, 1992. MR1219310 (94e:57015) Zbl 0768.51018
- [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