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− | * | + | Both Gromov's and Thurston's ''proof'' of this result make use of an averaging process. More precisely: |
+ | *''Gromov's strategy'': Use the duality principle (Corollary \ref{cor:svdual}) and average (bounded) continuous singular cochains over the isometry group of the Riemannian universal covering modulo the fundamental group; this requires a careful analysis of the relation between (bounded) continuous singular cohomology and (bounded) singular cohomology. | ||
+ | *''Thurston's strategy'': Replace singular homology by measure homology, and average measure chains over the the isometry group of the Riemannian universal covering; this requires a careful analysis of the relation between measure homology and singular homology. | ||
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Revision as of 15:04, 8 June 2010
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Contents |
1 Definition and history
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 and inheritance properties of the simplicial volume 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][Besson&Courtois&Gallot1991]:
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
Conversely, in the presence of negative curvature, the simplicial volume is bounded from below by the Riemannian volume [Gromov1982][Thurston1978, Theorem 6.2][Inoue&Yano1982]:
Theorem (Simplicial volume and negative sectional curvature) 3.2.
- The simplicial volume of oriented closed connected Riemannian manifolds of negative sectional curvature is non-zero. More precisely: For every there is a constant such that the following holds: If is an oriented closed connected Riemannian -manifold whose sectional curvature is bounded from above by , then
- Let be an oriented closed connected hyperbolic -manifold. Then , where is the supremal volume of all geodesic -simplices in hyperbolic -space (indeed, is finite [Thurston1978, Proposition 6.1.4]).
It is well known that [Thurston1978, p. 6.3], and hence, for any oriented closed connected surface of genus we have .
More generally, there are non-vanishing results for certain manifolds with negatively curved fundamental group.
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) 3.3. Let be a topological space, let , and let . Then
Corollary 3.4. 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:
Definition (Bounded cohomology) 3.5. 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 [Brooks1978], Ivanov [Ivanov1985], Noskov [Noskov1990], Monod [Monod2001][Monod2006], and Bühler [Bühler2008].
In the context of simplicial volume, bounded cohomology contributed to establish vanishing results in the presence of amenable fundamental groups, non-vanishing results in the presence of certain types of negative curvature, and inheritance properties with respect to products, connected sums, shared Riemannian coverings.
4 Inheritance properties
1 Products
The simplicial volume is almost multiplicative with respect to direct products of manifolds [Gromov1982, p. 10][Benedetti&Petronio1992, Theorem F.2.5]:
Theorem (Simplicial volume and products) 4.1. Let and be oriented closed connected manifolds. Then
A proof of the right hand estimate can be given by looking at the concrete description of in terms of the cross-product of singular chains; a proof of the left hand estimate can be obtained by using the duality principle (Corollary 3.4) and the fact that the norm is submultiplicative with respect to the cross-product of (bounded) singular cochains.
Notice that the simplicial volume in general is not multiplicative: Bucher-Karlsson [Bucher-Karlsson2008, Corollary 2] proved that holds for all oriented closed connected surfaces , of genus at least (and -- see above).
2 Connected sums
The simplicial volume is additive with respect to connected sums in the following sense [Gromov1982, p. 10]:
Theorem (Simplicial volume and connected sums) 4.2. Let and be oriented closed connected manifolds of dimension at least . Then
Notice that the simplicial volume in general is not additive with respect to connected sums in dimension : The simplicial volume of the torus is zero (see above), but the simplicial volume of an oriented closed connected surface of genus is non-zero (see above) is non-zero.
The proof of Theorem 4.2 is based on the mapping theorem in bounded cohomology (Theorem 5.1) and a careful analysis of so-called tree-like complexes [Gromov1982, Section 3.5]. Generalising these arguments, it can be seen that also additivity for the simplicial volume with respect to certain "amenable" glueings holds [Kuessner2001].
3 Fibrations
In general, the simplicial volume of a fibre bundle of oriented closed connected manifolds is not related in an obvious way to the simplicial volume of base and fibre:
- There exist oriented closed connected hyperbolic -manifolds that fibre over the circle. However, the circle has simplicial volume equal to zero, while the simplicial volume of the hyperbolic -manifold in question has non-zero simplicial volume.
- ...
4 Proportionality principle
For hyperbolic manifolds the simplicial volume is proportional to the Riemannian volume. Gromov and Thurston generalised this result suitably to cover all Riemannian manifolds:
Theorem (Proportionality principle) 4.3. Let and be oriented closed connected Riemannian manifolds that have isometric Riemannian universal coverings. Then
Both Gromov's and Thurston's proof of this result make use of an averaging process. More precisely:
- Gromov's strategy: Use the duality principle (Corollary 3.4) and average (bounded) continuous singular cochains over the isometry group of the Riemannian universal covering modulo the fundamental group; this requires a careful analysis of the relation between (bounded) continuous singular cohomology and (bounded) singular cohomology.
- Thurston's strategy: Replace singular homology by measure homology, and average measure chains over the the isometry group of the Riemannian universal covering; this requires a careful analysis of the relation between measure homology and singular homology.
5 Simplicial volume and the fundamental group
1 Background: Mapping theorem in bounded cohomology
Theorem (Mapping theorem in bounded cohomology) 5.1.
2 Amenability -- Vanishing results
3 Hyperbolicity -- Non-vanishing results
6 Applications
1 Mostow rigidity
2 Degree theorems
3 Bounded cohomology
7 References
- [Benedetti&Petronio1992] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Springer-Verlag, Berlin, 1992. MR1219310 (94e:57015) Zbl 0768.51018
- [Besson&Courtois&Gallot1991] G. Besson, G. Courtois and S. Gallot, Volume et entropie minimale des espaces localement symétriques, Invent. Math. 103 (1991), no.2, 417–445. MR1085114 (92d:58027) Zbl 0723.53029
- [Brooks1978] R. Brooks, Some remarks on bounded cohomology, in Riemann surfaces and related topics: Proceedings of the 1978 Stonybrook Conference, Ann. of Math. Stud., 97 (1978), 53–63. MR624804 (83a:57038) Zbl 0457.55002
- [Bucher-Karlsson2008] M. Bucher-Karlsson, The simplicial volume of closed manifolds covered by , J. Topol. 1 (2008), no.3, 584–602. MR2417444 (2009i:53025) Zbl 1156.53018
- [Bühler2008] T. Bühler, A derived functor approach to bounded cohomology, C. R. Math. Acad. Sci. Paris 346 (2008), no.11-12, 615–618. MR2423264 (2009f:18013) Zbl 1148.18007
- [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
- [Inoue&Yano1982] H. Inoue and K. Yano, The Gromov invariant of negatively curved manifolds, Topology 21 (1982), no.1, 83–89. MR630882 (82k:53091) Zbl 0469.53038
- [Ivanov1985] N. V. Ivanov, Foundations of the theory of bounded cohomology, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 143 (1985), 69–109, 177. MR806562 (87b:53070) Zbl 0612.55006
- [Kuessner2001] T. Kuessner, Relative simplicial volume, PhD thesis, Tübingen, 2001.
- [Monod2001] N. Monod, Continuous bounded cohomology of locally compact groups, Springer-Verlag, Berlin, 2001. MR1840942 (2002h:46121) Zbl 0967.22006
- [Monod2006] N. Monod, An invitation to bounded cohomology, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, (2006), 1183–1211. MR2275641 (2008e:22011) Zbl 1127.55002
- [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
- [Noskov1990] G. A. Noskov, Bounded cohomology of discrete groups with coefficients, Algebra i Analiz 2 (1990), no.5, 146–164. MR1086449 (92b:57005) Zbl 0729.55005
- [Thurston1978] W. P. Thurston, The Geometry and Topology of 3-Manifolds. Lecture notes, Princeton, 1978.
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 and inheritance properties of the simplicial volume 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][Besson&Courtois&Gallot1991]:
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
Conversely, in the presence of negative curvature, the simplicial volume is bounded from below by the Riemannian volume [Gromov1982][Thurston1978, Theorem 6.2][Inoue&Yano1982]:
Theorem (Simplicial volume and negative sectional curvature) 3.2.
- The simplicial volume of oriented closed connected Riemannian manifolds of negative sectional curvature is non-zero. More precisely: For every there is a constant such that the following holds: If is an oriented closed connected Riemannian -manifold whose sectional curvature is bounded from above by , then
- Let be an oriented closed connected hyperbolic -manifold. Then , where is the supremal volume of all geodesic -simplices in hyperbolic -space (indeed, is finite [Thurston1978, Proposition 6.1.4]).
It is well known that [Thurston1978, p. 6.3], and hence, for any oriented closed connected surface of genus we have .
More generally, there are non-vanishing results for certain manifolds with negatively curved fundamental group.
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) 3.3. Let be a topological space, let , and let . Then
Corollary 3.4. 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:
Definition (Bounded cohomology) 3.5. 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 [Brooks1978], Ivanov [Ivanov1985], Noskov [Noskov1990], Monod [Monod2001][Monod2006], and Bühler [Bühler2008].
In the context of simplicial volume, bounded cohomology contributed to establish vanishing results in the presence of amenable fundamental groups, non-vanishing results in the presence of certain types of negative curvature, and inheritance properties with respect to products, connected sums, shared Riemannian coverings.
4 Inheritance properties
1 Products
The simplicial volume is almost multiplicative with respect to direct products of manifolds [Gromov1982, p. 10][Benedetti&Petronio1992, Theorem F.2.5]:
Theorem (Simplicial volume and products) 4.1. Let and be oriented closed connected manifolds. Then
A proof of the right hand estimate can be given by looking at the concrete description of in terms of the cross-product of singular chains; a proof of the left hand estimate can be obtained by using the duality principle (Corollary 3.4) and the fact that the norm is submultiplicative with respect to the cross-product of (bounded) singular cochains.
Notice that the simplicial volume in general is not multiplicative: Bucher-Karlsson [Bucher-Karlsson2008, Corollary 2] proved that holds for all oriented closed connected surfaces , of genus at least (and -- see above).
2 Connected sums
The simplicial volume is additive with respect to connected sums in the following sense [Gromov1982, p. 10]:
Theorem (Simplicial volume and connected sums) 4.2. Let and be oriented closed connected manifolds of dimension at least . Then
Notice that the simplicial volume in general is not additive with respect to connected sums in dimension : The simplicial volume of the torus is zero (see above), but the simplicial volume of an oriented closed connected surface of genus is non-zero (see above) is non-zero.
The proof of Theorem 4.2 is based on the mapping theorem in bounded cohomology (Theorem 5.1) and a careful analysis of so-called tree-like complexes [Gromov1982, Section 3.5]. Generalising these arguments, it can be seen that also additivity for the simplicial volume with respect to certain "amenable" glueings holds [Kuessner2001].
3 Fibrations
In general, the simplicial volume of a fibre bundle of oriented closed connected manifolds is not related in an obvious way to the simplicial volume of base and fibre:
- There exist oriented closed connected hyperbolic -manifolds that fibre over the circle. However, the circle has simplicial volume equal to zero, while the simplicial volume of the hyperbolic -manifold in question has non-zero simplicial volume.
- ...
4 Proportionality principle
For hyperbolic manifolds the simplicial volume is proportional to the Riemannian volume. Gromov and Thurston generalised this result suitably to cover all Riemannian manifolds:
Theorem (Proportionality principle) 4.3. Let and be oriented closed connected Riemannian manifolds that have isometric Riemannian universal coverings. Then
Both Gromov's and Thurston's proof of this result make use of an averaging process. More precisely:
- Gromov's strategy: Use the duality principle (Corollary 3.4) and average (bounded) continuous singular cochains over the isometry group of the Riemannian universal covering modulo the fundamental group; this requires a careful analysis of the relation between (bounded) continuous singular cohomology and (bounded) singular cohomology.
- Thurston's strategy: Replace singular homology by measure homology, and average measure chains over the the isometry group of the Riemannian universal covering; this requires a careful analysis of the relation between measure homology and singular homology.
5 Simplicial volume and the fundamental group
1 Background: Mapping theorem in bounded cohomology
Theorem (Mapping theorem in bounded cohomology) 5.1.
2 Amenability -- Vanishing results
3 Hyperbolicity -- Non-vanishing results
6 Applications
1 Mostow rigidity
2 Degree theorems
3 Bounded cohomology
7 References
- [Benedetti&Petronio1992] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Springer-Verlag, Berlin, 1992. MR1219310 (94e:57015) Zbl 0768.51018
- [Besson&Courtois&Gallot1991] G. Besson, G. Courtois and S. Gallot, Volume et entropie minimale des espaces localement symétriques, Invent. Math. 103 (1991), no.2, 417–445. MR1085114 (92d:58027) Zbl 0723.53029
- [Brooks1978] R. Brooks, Some remarks on bounded cohomology, in Riemann surfaces and related topics: Proceedings of the 1978 Stonybrook Conference, Ann. of Math. Stud., 97 (1978), 53–63. MR624804 (83a:57038) Zbl 0457.55002
- [Bucher-Karlsson2008] M. Bucher-Karlsson, The simplicial volume of closed manifolds covered by , J. Topol. 1 (2008), no.3, 584–602. MR2417444 (2009i:53025) Zbl 1156.53018
- [Bühler2008] T. Bühler, A derived functor approach to bounded cohomology, C. R. Math. Acad. Sci. Paris 346 (2008), no.11-12, 615–618. MR2423264 (2009f:18013) Zbl 1148.18007
- [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
- [Inoue&Yano1982] H. Inoue and K. Yano, The Gromov invariant of negatively curved manifolds, Topology 21 (1982), no.1, 83–89. MR630882 (82k:53091) Zbl 0469.53038
- [Ivanov1985] N. V. Ivanov, Foundations of the theory of bounded cohomology, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 143 (1985), 69–109, 177. MR806562 (87b:53070) Zbl 0612.55006
- [Kuessner2001] T. Kuessner, Relative simplicial volume, PhD thesis, Tübingen, 2001.
- [Monod2001] N. Monod, Continuous bounded cohomology of locally compact groups, Springer-Verlag, Berlin, 2001. MR1840942 (2002h:46121) Zbl 0967.22006
- [Monod2006] N. Monod, An invitation to bounded cohomology, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, (2006), 1183–1211. MR2275641 (2008e:22011) Zbl 1127.55002
- [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
- [Noskov1990] G. A. Noskov, Bounded cohomology of discrete groups with coefficients, Algebra i Analiz 2 (1990), no.5, 146–164. MR1086449 (92b:57005) Zbl 0729.55005
- [Thurston1978] W. P. Thurston, The Geometry and Topology of 3-Manifolds. Lecture notes, Princeton, 1978.
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 and inheritance properties of the simplicial volume 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][Besson&Courtois&Gallot1991]:
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
Conversely, in the presence of negative curvature, the simplicial volume is bounded from below by the Riemannian volume [Gromov1982][Thurston1978, Theorem 6.2][Inoue&Yano1982]:
Theorem (Simplicial volume and negative sectional curvature) 3.2.
- The simplicial volume of oriented closed connected Riemannian manifolds of negative sectional curvature is non-zero. More precisely: For every there is a constant such that the following holds: If is an oriented closed connected Riemannian -manifold whose sectional curvature is bounded from above by , then
- Let be an oriented closed connected hyperbolic -manifold. Then , where is the supremal volume of all geodesic -simplices in hyperbolic -space (indeed, is finite [Thurston1978, Proposition 6.1.4]).
It is well known that [Thurston1978, p. 6.3], and hence, for any oriented closed connected surface of genus we have .
More generally, there are non-vanishing results for certain manifolds with negatively curved fundamental group.
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) 3.3. Let be a topological space, let , and let . Then
Corollary 3.4. 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:
Definition (Bounded cohomology) 3.5. 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 [Brooks1978], Ivanov [Ivanov1985], Noskov [Noskov1990], Monod [Monod2001][Monod2006], and Bühler [Bühler2008].
In the context of simplicial volume, bounded cohomology contributed to establish vanishing results in the presence of amenable fundamental groups, non-vanishing results in the presence of certain types of negative curvature, and inheritance properties with respect to products, connected sums, shared Riemannian coverings.
4 Inheritance properties
1 Products
The simplicial volume is almost multiplicative with respect to direct products of manifolds [Gromov1982, p. 10][Benedetti&Petronio1992, Theorem F.2.5]:
Theorem (Simplicial volume and products) 4.1. Let and be oriented closed connected manifolds. Then
A proof of the right hand estimate can be given by looking at the concrete description of in terms of the cross-product of singular chains; a proof of the left hand estimate can be obtained by using the duality principle (Corollary 3.4) and the fact that the norm is submultiplicative with respect to the cross-product of (bounded) singular cochains.
Notice that the simplicial volume in general is not multiplicative: Bucher-Karlsson [Bucher-Karlsson2008, Corollary 2] proved that holds for all oriented closed connected surfaces , of genus at least (and -- see above).
2 Connected sums
The simplicial volume is additive with respect to connected sums in the following sense [Gromov1982, p. 10]:
Theorem (Simplicial volume and connected sums) 4.2. Let and be oriented closed connected manifolds of dimension at least . Then
Notice that the simplicial volume in general is not additive with respect to connected sums in dimension : The simplicial volume of the torus is zero (see above), but the simplicial volume of an oriented closed connected surface of genus is non-zero (see above) is non-zero.
The proof of Theorem 4.2 is based on the mapping theorem in bounded cohomology (Theorem 5.1) and a careful analysis of so-called tree-like complexes [Gromov1982, Section 3.5]. Generalising these arguments, it can be seen that also additivity for the simplicial volume with respect to certain "amenable" glueings holds [Kuessner2001].
3 Fibrations
In general, the simplicial volume of a fibre bundle of oriented closed connected manifolds is not related in an obvious way to the simplicial volume of base and fibre:
- There exist oriented closed connected hyperbolic -manifolds that fibre over the circle. However, the circle has simplicial volume equal to zero, while the simplicial volume of the hyperbolic -manifold in question has non-zero simplicial volume.
- ...
4 Proportionality principle
For hyperbolic manifolds the simplicial volume is proportional to the Riemannian volume. Gromov and Thurston generalised this result suitably to cover all Riemannian manifolds:
Theorem (Proportionality principle) 4.3. Let and be oriented closed connected Riemannian manifolds that have isometric Riemannian universal coverings. Then
Both Gromov's and Thurston's proof of this result make use of an averaging process. More precisely:
- Gromov's strategy: Use the duality principle (Corollary 3.4) and average (bounded) continuous singular cochains over the isometry group of the Riemannian universal covering modulo the fundamental group; this requires a careful analysis of the relation between (bounded) continuous singular cohomology and (bounded) singular cohomology.
- Thurston's strategy: Replace singular homology by measure homology, and average measure chains over the the isometry group of the Riemannian universal covering; this requires a careful analysis of the relation between measure homology and singular homology.
5 Simplicial volume and the fundamental group
1 Background: Mapping theorem in bounded cohomology
Theorem (Mapping theorem in bounded cohomology) 5.1.
2 Amenability -- Vanishing results
3 Hyperbolicity -- Non-vanishing results
6 Applications
1 Mostow rigidity
2 Degree theorems
3 Bounded cohomology
7 References
- [Benedetti&Petronio1992] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Springer-Verlag, Berlin, 1992. MR1219310 (94e:57015) Zbl 0768.51018
- [Besson&Courtois&Gallot1991] G. Besson, G. Courtois and S. Gallot, Volume et entropie minimale des espaces localement symétriques, Invent. Math. 103 (1991), no.2, 417–445. MR1085114 (92d:58027) Zbl 0723.53029
- [Brooks1978] R. Brooks, Some remarks on bounded cohomology, in Riemann surfaces and related topics: Proceedings of the 1978 Stonybrook Conference, Ann. of Math. Stud., 97 (1978), 53–63. MR624804 (83a:57038) Zbl 0457.55002
- [Bucher-Karlsson2008] M. Bucher-Karlsson, The simplicial volume of closed manifolds covered by , J. Topol. 1 (2008), no.3, 584–602. MR2417444 (2009i:53025) Zbl 1156.53018
- [Bühler2008] T. Bühler, A derived functor approach to bounded cohomology, C. R. Math. Acad. Sci. Paris 346 (2008), no.11-12, 615–618. MR2423264 (2009f:18013) Zbl 1148.18007
- [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
- [Inoue&Yano1982] H. Inoue and K. Yano, The Gromov invariant of negatively curved manifolds, Topology 21 (1982), no.1, 83–89. MR630882 (82k:53091) Zbl 0469.53038
- [Ivanov1985] N. V. Ivanov, Foundations of the theory of bounded cohomology, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 143 (1985), 69–109, 177. MR806562 (87b:53070) Zbl 0612.55006
- [Kuessner2001] T. Kuessner, Relative simplicial volume, PhD thesis, Tübingen, 2001.
- [Monod2001] N. Monod, Continuous bounded cohomology of locally compact groups, Springer-Verlag, Berlin, 2001. MR1840942 (2002h:46121) Zbl 0967.22006
- [Monod2006] N. Monod, An invitation to bounded cohomology, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, (2006), 1183–1211. MR2275641 (2008e:22011) Zbl 1127.55002
- [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
- [Noskov1990] G. A. Noskov, Bounded cohomology of discrete groups with coefficients, Algebra i Analiz 2 (1990), no.5, 146–164. MR1086449 (92b:57005) Zbl 0729.55005
- [Thurston1978] W. P. Thurston, The Geometry and Topology of 3-Manifolds. Lecture notes, Princeton, 1978.
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- 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'''.
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