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 (Functoriality of the -semi-norm) 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 "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
![\displaystyle \|M\| \leq (n-1)^n \cdot n! \cdot \mathop{\mathrm{minvol}}(M).](/images/math/9/3/3/9333b5b2b3f72e9894f139f9d8d427f9.png)
The minimal volume [Gromov1982] of a complete smooth manifold is defined as
![\displaystyle \mathop{\mathrm{minvol}}(M) := \inf \bigl\{ \vol(M,g) \bigm| \text{$g$ is a Riemannian metric on~$M$ with~$|\mathop{\mathrm{sec}}(g)| \leq 1$}\bigr\}.](/images/math/9/4/f/94f298c07ed2fec614eeeb3c02cc2588.png)
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
![\displaystyle \|M\| > C_n \cdot |\delta|^{n/2} \cdot \vol(M).](/images/math/0/5/d/05dc3a9ef3cecd30ad7520ca53c3d0d5.png)
- 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
![\displaystyle \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}](/images/math/6/f/8/6f82251d9aa10ee21cd8647526da12dc.png)
Corollary 3.4.
Let be an oriented closed connected
-manifold. Then, where
denotes the cohomology class dual to the real fundamental class of
:
![\displaystyle \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}](/images/math/8/f/8/8f8573d5a0a0e74b35e344edeb32e5d8.png)
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
![\displaystyle \|\varphi\|_\infty := \bigl\{ |f|_\infty \bigm| \text{$f \in C^n_b(X;\mathbb{R})$ is a cocycle representing~$\varphi$} \bigr\}.](/images/math/0/d/6/0d6753af39191c27d1dac2e719d86c90.png)
- 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
[Gromov1982, p. 10]
Theorem (Simplicial volume and products) 4.1.
Let and
be oriented closed connected manifolds. Then
![\displaystyle \|M\| \cdot \|N\| \leq \| M \times N\| \leq {\dim(M) + \dim(N) \choose \dim(M)} \cdot \|M\| \cdot \|N\|.](/images/math/e/5/4/e54c7daa26fc75e73f9863f65be8ea36.png)
Notice that the simplicial volume in general is not multiplicative: Bucher-Karlsson [Bucher-Karlsson, ???] proved that
![\displaystyle \| S \times S'\| = ?? \cdot \|S\| \cdot \|S'\|](/images/math/6/1/8/6187d112570a682b2795dd0797c08ab8.png)
holds for all oriented closed connected surfaces ,
of genus at least
(and
-- see above).
2 Connected sums
[Gromov1982, p. 10]
Theorem (Simplicial volume and connected sums) 4.2.
Let and
be oriented closed connected manifolds of dimension at least
. Then
![\displaystyle \| M \mathbin{\#} N \| = \|M\| + \|N\|.](/images/math/8/4/0/840aba6b42448b9355fbdfee2cfc9aa8.png)
3 Fibrations
5 References
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- [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-Karlsson] Template:Bucher-Karlsson
- [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
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- [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
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- [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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