Simplicial volume

A simplicial volume is a topological invariant defined for closed manifolds . First considered by Gromov . Simplicial manifold volume $M$ ${\ displaystyle M}$ $M$ usually indicated $\|M\|$ ${\ displaystyle \ | M \ |}$ ${\ displaystyle \ | M \ |}$ .

Definition

Let be $M$ ${\ displaystyle M}$ Is a closed manifold, then

\|M\|=\inf \sum _{i}|r_{i}|

{\ displaystyle \ | M \ | = \ inf \ sum _ {i} | r_ {i} |}

,

Where $r_{i}$ ${\ displaystyle r_ {i}}$ - rational coefficients in the representation of its fundamental class $[M]$ ${\ displaystyle [M]}$ in terms of the sum of singular simplexes.

[M]=\sum _{i}r_{i}\Delta _{i}.

{\ displaystyle [M] = \ sum _ {i} r_ {i} \ Delta _ {i}.}

Properties

Gromov's theorem: The simplicial volume of a manifold of constant negative curvature is equal to the ratio of its volume to the volume of a regular infinite simplex in a Lobachevsky space of the same curvature.
For any varieties $M$ ${\ displaystyle M}$ and $N$ ${\ displaystyle N}$ same dimension
$\|M\#N\|=\|M\|+\|N\|$ ${\ displaystyle \ | M \ #N \ | = \ | M \ | + \ | N \ |}$ ,

Where

\#

{\ displaystyle \ #}

denotes a connected amount .

There are positive numbers $a(m)$ ${\ displaystyle a (m)}$ and $b(m)$ ${\ displaystyle b (m)}$ such that if the sum of dimensions $\dim M+\dim N\leqslant m$ ${\ displaystyle \ dim M + \ dim N \ leqslant m}$ then
$a(m)\|M\|\times \|N\|\leqslant \|M\times N\|\leqslant b(m)\|M\|\times \|N\|$ ${\ displaystyle a (m) \ | M \ | \ times \ | N \ | \ leqslant \ | M \ times N \ | \ leqslant b (m) \ | M \ | \ times \ | N \ |}$ ,

Where

\times

{\ displaystyle \ times}

denotes a direct work .

For any display $f$ $:$ $M$ $\to$ $N$ {\ displaystyle f \ colon M \ to N}
$\|M\|\geqslant (\deg f)\|N\|$ ${\ displaystyle \ | M \ | \ geqslant (\ deg f) \ | N \ |}$ where $\deg f$ ${\ displaystyle \ deg f}$ indicates the degree of display $f$ ${\ displaystyle f}$ . In particular:
- If the variety $M$ ${\ displaystyle M}$ allows display $M\to M$ ${\ displaystyle M \ to M}$ degrees of $>1$ ${\ displaystyle> 1}$ then $\|M\|=0$ ${\ displaystyle \ | M \ | = 0}$ .
- For anyone $n\geqslant 1$ ${\ displaystyle n \ geqslant 1}$ simplicial volume $n$ ${\ displaystyle n}$ -dimensional sphere equals $0$ ${\ displaystyle 0}$ .

Simplicial volume

Definition

Properties

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