Simplicial volume

Definition 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}} 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> <!-- COMMENT: 2do! rephrase defintion in terms of the l^1-semi-norm on homology add references complete the following sections add intro/history == Elementary examples == === "Functoriality" === <wikitex>; </wikitex> === Elementary examples === == "Computing" Simplicial volume == === Simplicial volume and Riemannian geometry === === Simplicial volume and bounded cohomology === == Further examples and properties == === Inheritance properties === ==== Products ==== ==== Connected sums ==== ==== Fibrations ==== === Vanishing results === === Non-vanishing results === == Applications == === Mostow rigidity === === Degree theorems === === Bounded cohomology === == Variations of the simplicial volume == === Simplicial volume of non-orientable manifolds === === Simplicial volume of non-compact manifolds === == Open problems == END OF COMMENT --> == References == {{#RefList:}} [[Category:Theory]] {{Stub}}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

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

Proposition 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.