Lace (mathematics)

In Mathematical, a lace is the modeling of a “loop”. It is a continuous and closed curve , i.e. its ends are confused. The concept of lace is useful in Analyze complexes and in Topologie.

Definitions

If $X$ is a topological Espace, one calls lace on $X$ all application continues $\backslash \; gamma\; \backslash :\; \backslash ,\; \backslash \; rightarrow\; X$ such as $\backslash \; gamma\; (0)\; =\; \backslash \; gamma\; (1)$.

Other definitions:

• a lace on $X$ is a way on $X$ whose end is confused with the origin.
• a lace on $X$ is a continuous application of $S^1$ towards $X$ (where $S^1$ indicates the Cercle unit $\backslash \; \{Z\; \backslash \; in\; \backslash \; mathbb\; \{C\}\; \backslash \; mid\; |Z|=1\; \backslash \}$).

In Analyze one complexes is interested in the laces which are also curved rectifiable.

One can also define polygonal laces, or of class $C^k$ (see Chemins).

Index of a lace in the Plane complex

See also: Index (analyzes complex)

In the case $X=\; \backslash \; mathbb\; \{C\}$, one can define the index $\backslash \; mathrm\; \{I\}\; (\backslash \; gamma,\; z\_0)$ of a lace $\backslash \; gamma$ compared to a point $z\_0\; \backslash \; in\; \backslash \; mathbb\; \{C\}\; \backslash \; smallsetminus\; \backslash \; gamma\; (1)$: it corresponds to the number (algebraic Entier) of turns carried out by the lace around this point.

One can obtain it while calculating:

See too

• Homotopy

• Connexity by arcs
• simple Connexity
• fundamental Group

• Analyze complexes | integral Theorem of Cauchy | Remainder theorem