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 $$\|M\| := \inf \bigl\{ \|c\|_1 \bigm| \text{$c \in C_n(M;\mathbb{R})$ is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0}. $$ Here, $C_*(M;\mathbb{R})$ denotes the singular chain complex of $M$ with real coefficients, and $\|\cdot\|_1$ denotes the $\ell^1$-norm on the singular chain complex induced from the (unordered) basis given by all singular simplices; i.e., for a chain $c=\sum_{j=0}^k a_j \cdot \sigma_j \in C_*(M;\mathbb{R})$ (in reduced form), the $\ell^1$-norm of $c$ is given by $$\|c\|_1 := \sum_{j=0}^k |a_j|.$$ {{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

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

Here, $C_*(M;\mathbb{R})$ denotes the singular chain complex of $M$ with real coefficients, and $\|\cdot\|_1$ denotes the $\ell^1$-norm on the singular chain complex induced from the (unordered) basis given by all singular simplices; i.e., for a chain $c=\sum_{j=0}^k a_j \cdot \sigma_j \in C_*(M;\mathbb{R})$ (in reduced form), the $\ell^1$-norm of $c$ is given by

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