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 1.1. Let $M$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. DON'T FORGET TO ENTER YOUR USER NAME INTO THE {{Authors| }} TEMPLATE BELOW. END OF COMMENT -->{{Authors|Clara Löh}} == Definition and history == <wikitex>; The ''simplicial volume'' is a homotopy invariant of oriented closed connected manifolds that was introduced by Gromov in his proof of Mostow rigidity {{cite|Munkholm1980}}{{cite|Gromov1982}}. Intuitively, the simplicial volume measures how difficult it is to describe the manifold in question in terms of simplices (with real coefficients): {{Beginthm|Definition}} Let $M$ be an oriented closed connected manifold of dimension $n$. Then the '''simplicial volume''' (also called '''Gromov norm''') of $M $ is defined as $$\|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\}.$$ {{Endthm}} </wikitex> == Functoriality and elementary examples == <wikitex> The $\ell^1$-semi-norm is functorial in the following sense {{cite|Gromov1999}}: {{Beginthm|Proposition}} If $f \colon X \longrightarrow Y$ is a continuous map of topological spaces and $\alpha \in H_*(X;\mathbb{R})$, then $$ \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})$$C_*(\,\cdot\,;\mathbb{R})$ with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space $X$$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 $X$$X$ is a topological space and $\alpha \in H_*(X;\mathbb{R})$$\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\}.$ $$\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$$

2 Functoriality and elementary examples

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

Proposition 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 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.

$, 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})$$C_*(\,\cdot\,;\mathbb{R})$ with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space $X$$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 $X$$X$ is a topological space and $\alpha \in H_*(X;\mathbb{R})$$\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\}.$ $$\displaystyle \|\alpha\|_1 := \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_*(X;\mathbb{R})$ is a cycle representing~$\alpha$}\bigr\}.$$