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 $M$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. DON'T FORGET TO ENTER YOUR USER NAME INTO THE {{Authors| }} TEMPLATE BELOW. END OF COMMENT -->{{Authors|Clara Löh}} == Definition and history == <wikitex>; The ''simplicial volume'' is a homotopy invariant of oriented closed connected manifolds that was introduced by Gromov in his proof of Mostow rigidity {{cite|Munkholm1980}}{{cite|Gromov1982}}. Intuitively, the simplicial volume measures how difficult it is to describe the manifold in question in terms of simplices (with real coefficients): {{Beginthm|Definition (Simplicial volume)}} Let $M$ be an oriented closed connected manifold of dimension $n$. Then the '''simplicial volume''' (also called '''Gromov norm''') of $M $ is defined as $$\|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}, $$ where $[M] \in H_n(M;\mathbb{R})$ is the fundamental class of $M$ with real coefficients. * Here, $|\cdot|_1$ denotes the $\ell^1$-norm on the singular chain complex $C_*(\,\cdot\,;\mathbb{R})$ with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space $X$ and a chain $c = \sum_{j=0}^{k} a_j \cdot \sigma_j \in C_*(X;\mathbb{R})$ (in reduced form), the '''$\ell^1$-norm''' of $c$ is given by $$ |c|_1 := \sum_{j=0}^k |a_j|.$$ * Moreover, $\|\cdot\|_1$ denotes the '''$\ell^1$-semi-norm''' on singular homology $H_*(\,\cdot\,;\mathbb{R})$ with real coefficients, which is induced by $|\cdot|_1$. More explicitly, if $X$ is a topological space and $\alpha \in H_*(X;\mathbb{R})$, then $$ \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$$ {{Endthm}} {{Beginthm|Convention}} In the following, if not explicitly stated otherwise, all manifolds are topological manifolds and are of non-zero dimension. {{Endthm}} </wikitex> == Functoriality and elementary examples == <wikitex> The $\ell^1$-semi-norm is functorial in the following sense {{cite|Gromov1999}}: {{Beginthm|Proposition (Functoriality of the $\ell^1$-semi-norm)}} If $f \colon X \longrightarrow Y$ is a continuous map of topological spaces and $\alpha \in H_*(X;\mathbb{R})$, then $$ \bigl\| H_*(f;\mathbb{R}) (\alpha) \bigr\|_1 \leq \|\alpha\|_1,$$ as can be seen by inspecting the definition of $H_*(f;\mathbb{R}) = H_*(C_*(f;\mathbb{R}))$ and of $\|\cdot\|_1$. {{Endthm}} {{Beginthm|Corollary}} * Let $f \colon M\longrightarrow N$ be a map of oriented closed connected manifolds of the same dimension. Then $$ |\deg f| \cdot \|N\| \leq \|M\|.$$ * Because homotopy equivalences of oriented closed connected manifolds have degree $-1$ or M$ be an oriented closed connected manifold of dimension $n$$n$. Then the simplicial volume (also called Gromov norm) of $M$$M$ 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},$ $$\displaystyle \|M\| := \bigl\| [M] \bigr\|_1 = \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_n(M;\mathbb{R})$ is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0},$$

where $[M] \in H_n(M;\mathbb{R})$$[M] \in H_n(M;\mathbb{R})$ is the fundamental class of $M$$M$ with real coefficients.

• Here, $|\cdot|_1$$|\cdot|_1$ denotes the $\ell^1$$\ell^1$-norm on the singular chain complex $C_*(\,\cdot\,;\mathbb{R})$$C_*(\,\cdot\,;\mathbb{R})$ with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space

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  $X$ and a chain $c = \sum_{j=0}^{k} a_j \cdot \sigma_j \in C_*(X;\mathbb{R})$$c = \sum_{j=0}^{k} a_j \cdot \sigma_j \in C_*(X;\mathbb{R})$ (in reduced form), the $\ell^1$$\ell^1$-norm of $c$$c$ is given by

$\displaystyle |c|_1 := \sum_{j=0}^k |a_j|.$ $$\displaystyle |c|_1 := \sum_{j=0}^k |a_j|.$$

• Moreover, $\|\cdot\|_1$$\|\cdot\|_1$ denotes the $\ell^1$$\ell^1$-semi-norm on singular homology $H_*(\,\cdot\,;\mathbb{R})$$H_*(\,\cdot\,;\mathbb{R})$ with real coefficients, which is induced by $|\cdot|_1$$|\cdot|_1$. More explicitly, if

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  $X$ is a topological space and $\alpha \in H_*(X;\mathbb{R})$$\alpha \in H_*(X;\mathbb{R})$, then

$\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$ $$\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$$

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 $\ell^1$$\ell^1$-semi-norm is functorial in the following sense [Gromov1999]:

Proposition (Functoriality of the $\ell^1$$\ell^1$-semi-norm) 2.1. If $f \colon X \longrightarrow Y$$f \colon X \longrightarrow Y$ is a continuous map of topological spaces and $\alpha \in H_*(X;\mathbb{R})$$\alpha \in H_*(X;\mathbb{R})$, then

$\displaystyle \bigl\| H_*(f;\mathbb{R}) (\alpha) \bigr\|_1 \leq \|\alpha\|_1,$ $$\displaystyle \bigl\| H_*(f;\mathbb{R}) (\alpha) \bigr\|_1 \leq \|\alpha\|_1,$$

as can be seen by inspecting the definition of $H_*(f;\mathbb{R}) = H_*(C_*(f;\mathbb{R}))$$H_*(f;\mathbb{R}) = H_*(C_*(f;\mathbb{R}))$ and of $\|\cdot\|_1$$\|\cdot\|_1$.

Corollary 2.2.

• Let $f \colon M\longrightarrow N$$f \colon M\longrightarrow N$ be a map of oriented closed connected manifolds of the same dimension. Then

$\displaystyle |\deg f| \cdot \|N\| \leq \|M\|.$ $$\displaystyle |\deg f| \cdot \|N\| \leq \|M\|.$$

• Because homotopy equivalences of oriented closed connected manifolds have degree $-1$$-1$ or $1$$1$, 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 $-1$$-1$, $0$$0$, or $1$$1$) 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 $n$$n$-manifolds $M$$M$ we have

$\displaystyle \|M\| \leq (n-1)^n \cdot n! \cdot \mathop{\mathrm{minvol}}(M).$ $$\displaystyle \|M\| \leq (n-1)^n \cdot n! \cdot \mathop{\mathrm{minvol}}(M).$$

The minimal volume [Gromov1982] of a complete smooth manifold $M$$M$ 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\}.$ $$\displaystyle \mathop{\mathrm{minvol}}(M) := \inf \bigl\{ \vol(M,g) \bigm| \text{$g$ is a Riemannian metric on~$M$ with~$|\mathop{\mathrm{sec}}(g)| \leq 1$}\bigr\}.$$

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

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  $n \in \mathbb{N}$ there is a constant $C_n \in \mathbb{R}_{>0}$$C_n \in \mathbb{R}_{>0}$ such that the following holds: If $M$$M$ is an oriented closed connected Riemannian $n$$n$-manifold whose sectional curvature is bounded from above by $\delta \in \mathbb{R}_{<0}$$\delta \in \mathbb{R}_{<0}$, then

$\displaystyle \|M\| > C_n \cdot |\delta|^{n/2} \cdot \vol(M).$ $$\displaystyle \|M\| > C_n \cdot |\delta|^{n/2} \cdot \vol(M).$$

• Let $M$$M$ be an oriented closed connected hyperbolic $n$$n$-manifold. Then $\|M\| = \vol(M)/v_n$$\|M\| = \vol(M)/v_n$, where $v_n$$v_n$ is the supremal volume of all geodesic $n$$n$-simplices in hyperbolic $n$$n$-space (indeed, $v_n$$v_n$ is finite [Thurston1978, Proposition 6.1.4]).

It is well known that $v_2=\pi$$v_2=\pi$ [Thurston1978, p. 6.3], and hence, for any oriented closed connected surface $S_g$$S_g$ of genus $g\in \mathbb{N}_{\geq 1}$$g\in \mathbb{N}_{\geq 1}$ we have $\|S_g\| = 4\cdot g - 4$$\|S_g\| = 4\cdot g - 4$.

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

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$X$ be a topological space, let

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$n \in \mathbb{N}$, and let

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$\alpha \in H_n(X;\mathbb{R})$. Then

$\displaystyle \begin{aligned} \|\alpha\|_1 &= \sup \Bigl\{ \frac{1}{\|\varphi\|_\infty} \Bigm| \varphi \in H^n(X;\mathbb{R}),~\langle \varphi, \alpha\rangle = 1 \Bigr\} \\ &= \sup \Bigl\{ \frac{1}{\|\varphi\|_\infty} \Bigm| \varphi \in H_b^n(X;\mathbb{R}),~\langle \varphi, \alpha\rangle = 1 \Bigr\}. \end{aligned}$ $$\displaystyle \begin{aligned} \|\alpha\|_1 &= \sup \Bigl\{ \frac{1}{\|\varphi\|_\infty} \Bigm| \varphi \in H^n(X;\mathbb{R}),~\langle \varphi, \alpha\rangle = 1 \Bigr\} \\ &= \sup \Bigl\{ \frac{1}{\|\varphi\|_\infty} \Bigm| \varphi \in H_b^n(X;\mathbb{R}),~\langle \varphi, \alpha\rangle = 1 \Bigr\}. \end{aligned}$$

Corollary 3.4.

Let $M$$M$ be an oriented closed connected $n$$n$-manifold. Then, where

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$[M]^* \in H^n(M;\mathbb{R})$ denotes the cohomology class dual to the real fundamental class of $M$$M$:

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$$\displaystyle \begin{aligned} \| M \| & = \frac{1}{\bigl\| [M]^* \bigr\|_\infty}\\ & = \sup \Bigl\{ \frac{1}{\|\varphi\|_\infty} \Bigm| \varphi \in H_b^n(M;\mathbb{R}),~c_M(\varphi) = [M]^* \Bigr\}. \end{aligned}$$

For the sake of completeness, we review the definition of bounded cohomology of topological spaces:

Definition (Bounded cohomology) 3.5.

Let

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$X$ be a topological space, and let

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$n \in \mathbb{N}$.

• If

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  $f \in C^n(X;\mathbb{R})$ is a cochain, then we write

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  $$\displaystyle |f|_\infty := \sup_{\sigma \in \mathop{\mathrm{map}} (\Delta^n,X)} |f(\sigma)| \in \mathbb{R}_{\geq 0} \cup \{\infty\}.$$
  If

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  $|f|_\infty < \infty$, then

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  $f$ is a bounded cochain.

• We write $C_b^n(X;\mathbb{R}) := \bigl\{ f \bigm| f \in C^n(X;\mathbb{R}),~|f|_\infty < \infty$$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

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  $C_b^*(X;\mathbb{R})$ is a subcomplex of the singular cochain complex, called the bounded cochain complex of

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  $X$.
• The cohomology

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  $H^*_b(X;\mathbb{R})$ of $C^*(X;\mathbb{R})$$C^*(X;\mathbb{R})$ is the bounded cohomology of

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  $X$.
• The norm

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  $|\cdot|_\infty$ on the bounded cochain complex induces a semi-norm on bounded cohomology: If

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  $\varphi \in H^n_b(X;\mathbb{R})$, then

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$$\displaystyle \|\varphi\|_\infty := \bigl\{ |f|_\infty \bigm| \text{$f \in C^n_b(X;\mathbb{R})$ is a cocycle representing~$\varphi$} \bigr\}.$$

• The inclusion

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  $C_b^*(X;\mathbb{R}) \hookrightarrow C^*(X;\mathbb{R})$ induces a homomorphism

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  $c_X \colon H^*_b(X;\mathbb{R}) \longrightarrow H^*(X;\mathbb{R})$, 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 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
• [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
• [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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$, it follows that the simplicial volume indeed is a homotopy invariant of oriented closed connected manifolds. {{Endthm}} Hence, all oriented closed connected manifolds admitting a self-map of non-trivial degree (i.e., not equal to $-1$, $M$ be an oriented closed connected manifold of dimension $n$$n$. Then the simplicial volume (also called Gromov norm) of $M$$M$ 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},$ $$\displaystyle \|M\| := \bigl\| [M] \bigr\|_1 = \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_n(M;\mathbb{R})$ is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0},$$

where $[M] \in H_n(M;\mathbb{R})$$[M] \in H_n(M;\mathbb{R})$ is the fundamental class of $M$$M$ with real coefficients.

• Here, $|\cdot|_1$$|\cdot|_1$ denotes the $\ell^1$$\ell^1$-norm on the singular chain complex $C_*(\,\cdot\,;\mathbb{R})$$C_*(\,\cdot\,;\mathbb{R})$ with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space

  Tex syntax error

  $X$ and a chain $c = \sum_{j=0}^{k} a_j \cdot \sigma_j \in C_*(X;\mathbb{R})$$c = \sum_{j=0}^{k} a_j \cdot \sigma_j \in C_*(X;\mathbb{R})$ (in reduced form), the $\ell^1$$\ell^1$-norm of $c$$c$ is given by

$\displaystyle |c|_1 := \sum_{j=0}^k |a_j|.$ $$\displaystyle |c|_1 := \sum_{j=0}^k |a_j|.$$

• Moreover, $\|\cdot\|_1$$\|\cdot\|_1$ denotes the $\ell^1$$\ell^1$-semi-norm on singular homology $H_*(\,\cdot\,;\mathbb{R})$$H_*(\,\cdot\,;\mathbb{R})$ with real coefficients, which is induced by $|\cdot|_1$$|\cdot|_1$. More explicitly, if

  Tex syntax error

  $X$ is a topological space and $\alpha \in H_*(X;\mathbb{R})$$\alpha \in H_*(X;\mathbb{R})$, then

$\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$ $$\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$$