Simplicial volume
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1 Definition and history
Definition 1.1.
Let be an oriented closed connected manifold of dimension
. Then the simplicial volume (also called Gromov norm) of
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}.](/images/math/c/1/4/c146c512897385d5bf7d07e0c5cdb0f9.png)
Here, denotes the singular chain complex of
with real coefficients, and
denotes the
-norm on the singular chain complex induced from the (unordered) basis given by all singular simplices; i.e., for a chain
(in reduced form), the
-norm of
is given by
![\displaystyle \|c\|_1 := \sum_{j=0}^k |a_j|.](/images/math/c/9/5/c95d57c1c01b7f966989ceac5084337f.png)
2 References
This page has not been refereed. The information given here might be incomplete or provisional. |