Simplicial volume

Definition (Simplicial volume) 1.1. Let $<!-- 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 $n$. Then the simplicial volume (also called Gromov norm) of $M$ is defined as

$$\displaystyle \|M\| := \bigl\| [M] \bigr\|_1 = \inf \bigl\{ |c|_1 \bigm| \text{$c \in C_n(M;\mathbb{R})$ is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0},$$

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

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

• 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

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

Convention 1.2. In the following, if not explicitly stated otherwise, all manifolds are topological manifolds and are of non-zero dimension.

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

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

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

Corollary 2.2.

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

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

• Because homotopy equivalences of oriented closed connected manifolds have degree $-1$ or $1$, it follows that the simplicial volume indeed is a homotopy invariant of oriented closed connected manifolds.

Theorem (Simplicial volume and minimal volume) 3.1. For all oriented closed connected smooth $n$-manifolds $M$ we have

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

Proposition (Duality principle) 3.2.

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

Corollary 3.3.

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Definition (Bounded cohomology) 3.4.

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• 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 $f$ is a bounded cochain.

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

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