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The branch of topology and algebra concerned with braids, the groups formed by their equivalence classes and various generalizations of these groups [[#References|[1]]].
 
The branch of topology and algebra concerned with braids, the groups formed by their equivalence classes and various generalizations of these groups [[#References|[1]]].
  
A braid on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b0174702.png" /> strings is an object consisting of two parallel planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b0174703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b0174704.png" /> in three-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b0174705.png" />, containing two ordered sets of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b0174706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b0174707.png" />, and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b0174708.png" /> simple non-intersecting arcs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b0174709.png" />, intersecting each parallel plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747010.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747012.png" /> exactly once and joining the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747015.png" />. It is assumed that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747016.png" />'s lie on a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747018.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747019.png" />'s on a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747021.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747022.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747023.png" /> lies beneath <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747024.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747025.png" /> (see Fig. a). Braids can be represented in the projection on the plane passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747027.png" />; this projection can be brought into general position in such a way that there are only finitely many double points, each two of which lie at different levels, and the intersections are transversal.
+
A braid on $  n $
 +
strings is an object consisting of two parallel planes $  P _ {0} $
 +
and $  P _ {1} $
 +
in three-dimensional space $  \mathbf R  ^ {3} $,  
 +
containing two ordered sets of points $  a _ {1} \dots a _ {n} \in P _ {0} $
 +
and  $  b _ {1} \dots b _ {n} \in P _ {1} $,  
 +
and of $  n $
 +
simple non-intersecting arcs $  l _ {1} \dots l _ {n} $,  
 +
intersecting each parallel plane $  P _ {t} $
 +
between $  P _ {0} $
 +
and $  P _ {1} $
 +
exactly once and joining the points $  \{ a _ {i} \} $
 +
to $  \{ b _ {i} \} $,  
 +
$  i = 1 \dots n $.  
 +
It is assumed that the $  a _ {i} $'
 +
s lie on a straight line $  L _ {a} $
 +
in $  P _ {0} $
 +
and the b _ {i} $'
 +
s on a straight line $  L _ {b} $
 +
in $  P _ {1} $
 +
parallel to $  L _ {a} $;  
 +
moreover, b _ {i} $
 +
lies beneath $  a _ {i} $
 +
for each $  i $(
 +
see Fig. a). Braids can be represented in the projection on the plane passing through $  L _ {a} $
 +
and $  L _ {b} $;  
 +
this projection can be brought into general position in such a way that there are only finitely many double points, each two of which lie at different levels, and the intersections are transversal.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017470a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017470a.gif" />
Line 11: Line 49:
 
Figure: b017470b
 
Figure: b017470b
  
The string <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747028.png" /> of a braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747029.png" /> joins <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747030.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747031.png" /> and so defines a permutation
+
The string $  l _ {i} $
 +
of a braid $  \omega $
 +
joins $  a _ {i} $
 +
to b _ {k _ {i}  } $
 +
and so defines a permutation
 +
 
 +
$$
 +
S  ^  \omega  = \
 +
\left ( \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747032.png" /></td> </tr></table>
+
\begin{array}{ccc}
 +
1  &\dots  & n  \\
 +
k _ {1}  &\dots  &k _ {n}  \\
 +
\end{array}
 +
\
 +
\right ) .
 +
$$
  
If this is the identity permutation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747033.png" /> is called a coloured (or pure) braid. The transposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747034.png" /> corresponds to a simple braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747035.png" /> (see Fig. b).
+
If this is the identity permutation, $  \omega $
 +
is called a coloured (or pure) braid. The transposition $  (i  i + 1) $
 +
corresponds to a simple braid $  \sigma _ {i} $(
 +
see Fig. b).
  
On the set of all braids on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747036.png" /> strings with fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747040.png" />, one introduces the equivalence relation defined by homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747042.png" /> is the region between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747044.png" />, which reduce to the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747045.png" />; it may be assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747046.png" />. Braids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747048.png" /> are equivalent if there exists a homeomorphism with the above properties such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747049.png" />.
+
On the set of all braids on $  n $
 +
strings with fixed $  P _ {0} $,  
 +
$  P _ {1} $,  
 +
$  \{ a _ {i} \} $,  
 +
$  \{ b _ {i} \} $,  
 +
one introduces the equivalence relation defined by homeomorphisms $  h: \Pi \rightarrow \Pi $,  
 +
where $  \Pi $
 +
is the region between $  P _ {0} $
 +
and $  P _ {1} $,  
 +
which reduce to the identity on $  P _ {0} \cup P _ {1} $;  
 +
it may be assumed that $  h (P _ {t} ) = P _ {t} $.  
 +
Braids $  \alpha $
 +
and $  \beta $
 +
are equivalent if there exists a homeomorphism with the above properties such that $  h ( \alpha ) = \beta $.
  
The equivalence classes — which are still called braids — form the braid group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747050.png" /> with respect to the operation defined as follows. Place a copy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747051.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747052.png" /> above another copy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747053.png" />, in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747054.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747056.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747057.png" />, and then compress <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747058.png" /> to half its "height" . The images of the braids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747060.png" /> produce a braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747061.png" />, with string <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747062.png" /> obtained by extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747063.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747065.png" />. The identity braid is the equivalence class containing the braid with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747066.png" /> parallel segments; the inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747067.png" /> of a braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747068.png" /> is defined by reflection in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747069.png" />. For the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747070.png" /> see Fig. c.
+
The equivalence classes — which are still called braids — form the braid group $  B (n) $
 +
with respect to the operation defined as follows. Place a copy $  \Pi  ^  \prime  $
 +
of the domain $  \Pi $
 +
above another copy $  \Pi  ^ {\prime\prime} $,  
 +
in such a way that $  P _ {0}  ^ {\prime\prime} $
 +
coincides with $  P _ {1}  ^  \prime  $,  
 +
$  a _ {i}  ^ {\prime\prime} $
 +
with b _ {i}  ^  \prime  $,  
 +
and then compress $  \Pi  ^  \prime  \cup \Pi  ^ {\prime\prime} $
 +
to half its "height" . The images of the braids $  \omega  ^  \prime  \in \Pi  ^  \prime  $
 +
and $  \omega  ^ {\prime\prime} \in \Pi  ^ {\prime\prime} $
 +
produce a braid $  \omega  ^  \prime  \omega  ^ {\prime\prime} $,  
 +
with string $  l _ {i} $
 +
obtained by extending $  l _ {i}  ^  \prime  $
 +
with $  l _ {k _ {i}  }  ^ {\prime\prime} $,  
 +
where $  k _ {i} \in S ^ {\omega  ^  \prime  (i) } $.  
 +
The identity braid is the equivalence class containing the braid with $  n $
 +
parallel segments; the inverse $  \omega  ^ {-1} $
 +
of a braid $  \omega $
 +
is defined by reflection in the plane $  P _ {1/2} $.  
 +
For the condition $  \omega \omega  ^ {-1} = \epsilon $
 +
see Fig. c.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017470c.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017470c.gif" />
Line 25: Line 114:
 
Figure: b017470c
 
Figure: b017470c
  
The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747071.png" /> defines an epimorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747072.png" /> onto the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747073.png" /> of permutations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747074.png" /> elements, the kernel of this epimorphism is the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747075.png" /> of all pure braids, so that one has an exact sequence
+
The mapping $  \omega \rightarrow S  ^  \omega  $
 +
defines an epimorphism of $  B (n) $
 +
onto the group $  S (n) $
 +
of permutations of $  n $
 +
elements, the kernel of this epimorphism is the subgroup $  K (n) $
 +
of all pure braids, so that one has an exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747076.png" /></td> </tr></table>
+
$$
 +
1  \rightarrow  K (n)  \rightarrow  B (n)  \rightarrow  S (n)  \rightarrow  1.
 +
$$
  
The braid group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747077.png" /> has two principal interpretations. The first, as a configuration space, is obtained by identifying the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747078.png" /> via vertical projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747079.png" />, under which the images of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747080.png" />, considered as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747081.png" /> varies from 0 to 1, form the trace of an isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747082.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747083.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747084.png" />; one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747085.png" />. Consider the space of unordered sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747086.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747087.png" /> pairwise distinct points of the plane; then each braid corresponds in one-to-one fashion to a class of homotopy loops in this space, and one has an isomorphism
+
The braid group $  B (n) $
 +
has two principal interpretations. The first, as a configuration space, is obtained by identifying the planes $  P _ {t} $
 +
via vertical projection onto $  P _ {0} $,  
 +
under which the images of the points $  a _ {it }  = l _ {i} \cap P _ {t} $,  
 +
considered as $  t $
 +
varies from 0 to 1, form the trace of an isotopy $  \phi _ {t}  ^  \omega  $
 +
of the set $  \cup a _ {i} $
 +
along $  P _ {0} $;  
 +
one has $  \phi _ {1}  ^  \omega  ( \cup a _ {i} ) = \cup a _ {i} $.  
 +
Consider the space of unordered sequences $  G (n) $
 +
of $  n $
 +
pairwise distinct points of the plane; then each braid corresponds in one-to-one fashion to a class of homotopy loops in this space, and one has an isomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747088.png" /></td> </tr></table>
+
$$
 +
\beta : B (n)  \rightarrow \
 +
\pi _ {1} G (n).
 +
$$
  
 
For pure braids one has an analogously constructed isomorphism
 
For pure braids one has an analogously constructed isomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747089.png" /></td> </tr></table>
+
$$
 +
\alpha : K (n)  \rightarrow \
 +
\pi _ {1} F (n),
 +
$$
 +
 
 +
where  $  F (n) $
 +
is the space of ordered sequences of  $  n $
 +
distinct points of the plane, so that  $  K (n) $
 +
can be identified with the subgroup corresponding to the covering
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747090.png" /> is the space of ordered sequences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747091.png" /> distinct points of the plane, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747092.png" /> can be identified with the subgroup corresponding to the covering
+
$$
 +
p: F (n)  \rightarrow  G (n)  = \
 +
F (n) / S (n).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747093.png" /></td> </tr></table>
+
The second interpretation, as a homeotopy group, is obtained by extending the isotopy  $  \phi _ {t}  ^  \omega  $
 +
to an isotopy  $  {\widetilde \phi  } {} _ {t}  ^  \omega  $
 +
of the plane  $  P _ {0} $
 +
that coincides with the identity outside some disc, and such that  $  {\widetilde \phi  } {} _ {0}  ^  \omega  = \mathop{\rm id} $.  
 +
For each  $  t $,
 +
two such extensions differ by a homeomorphism which is the identity at the points  $  a _ {it} $.  
 +
A braid uniquely determines a component of the space of homeomorphisms  $  Y (n) $
 +
of the plane which map the set  $  \cup a _ {i} $
 +
onto itself, and one has an isomorphism
  
The second interpretation, as a homeotopy group, is obtained by extending the isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747094.png" /> to an isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747095.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747096.png" /> that coincides with the identity outside some disc, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747097.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747098.png" />, two such extensions differ by a homeomorphism which is the identity at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b01747099.png" />. A braid uniquely determines a component of the space of homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470100.png" /> of the plane which map the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470101.png" /> onto itself, and one has an isomorphism
+
$$
 +
\gamma : B (n)  \rightarrow \
 +
\pi _ {0} Y (n).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470102.png" /></td> </tr></table>
+
To each homeomorphism  $  h \in Y $
 +
corresponds an automorphism of the free group of rank  $  n: $
 +
$  F _ {n} = \pi _ {1} ( \mathbf R  ^ {2} \setminus  \cup a _ {i} ) $,
 +
defined up to an inner automorphism, which in turn yields a homomorphism  $  B (n) \rightarrow  \mathop{\rm Out}  F _ {n} = \mathop{\rm Aut}  F _ {n} / \mathop{\rm Inn}  F _ {n} $.  
 +
The elements of the image are called braid automorphisms of the free group. In particular, corresponding to the braid  $  \sigma _ {i} $
 +
one has an automorphism
  
To each homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470103.png" /> corresponds an automorphism of the free group of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470104.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470105.png" />, defined up to an inner automorphism, which in turn yields a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470106.png" />. The elements of the image are called braid automorphisms of the free group. In particular, corresponding to the braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470107.png" /> one has an automorphism
+
$$
 +
\overline \sigma \; _ {i} :\
 +
\overline \sigma \; _ {i} (x _ {i} )  = \
 +
x _ {i + 1 }  ,\ \
 +
\overline \sigma \; _ {i} (x _ {i + 1 }  )  = \
 +
x _ {i + 1 }  x _ {i} x _ {i + 1 }  ^ {-1} ,\ \
 +
\overline \sigma \; _ {i} (x _ {j} )  = x _ {j} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470108.png" /></td> </tr></table>
+
if  $  j \neq i, i + 1 $(
 +
$  \{ x _ {i} \} $
 +
is a set of generators of  $  F _ {n} $).
 +
Any braid automorphism  $  \alpha $
 +
possesses the following properties:
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470109.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470110.png" /> is a set of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470111.png" />). Any braid automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470112.png" /> possesses the following properties:
+
$$
 +
\alpha (x _ {i} )  = \
 +
A _ {i} x _ {i} A _ {i}  ^ {-1} ,\ \
 +
\alpha \left ( \prod _ {i = 1 } ^ { n }  x _ {i} \right ) = \
 +
\prod _ {i = 1 } ^ { n }  x _ {i} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470113.png" /></td> </tr></table>
+
up to an inner automorphism (for the meaning of  $  A _ {i} $,
 +
see below); these properties characterize braid automorphisms.
  
up to an inner automorphism (for the meaning of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470114.png" />, see below); these properties characterize braid automorphisms.
+
The braids  $  \sigma _ {i} , 1 \leq  i \leq  n - 1 $,
 +
are the generators of the group  $  B (n) $,
 +
i.e. $  \omega = \sigma _ {k _ {1}  } \dots \sigma _ {k _ {m}  } $,  
 +
with
  
The braids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470115.png" />, are the generators of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470116.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470117.png" />, with
+
$$ \tag{1 }
 +
\left .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470118.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\begin{array}{ll}
 +
\sigma _ {i} \sigma _ {j}  = \
 +
\sigma _ {j} \sigma _ {i}  & \textrm{ if }  | i - j | > 1,  \\
 +
\sigma _ {i} \sigma _ {i + 1 }  \sigma _ {i}  = \
 +
\sigma _ {i + 1 }  \sigma _ {i} \sigma _ {i - 1 }  ,  & 1 \leq  i \leq  n - 2. \\
 +
\end{array}
 +
\
 +
\right \}
 +
$$
  
It turns out that (1) is a presentation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470119.png" /> (see Fig. d).
+
It turns out that (1) is a presentation of $  B (n) $(
 +
see Fig. d).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017470d.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017470d.gif" />
Line 69: Line 236:
 
Figure: b017470e
 
Figure: b017470e
  
There exists a splitting exact sequence (obtained from the locally trivial fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470120.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470121.png" />):
+
There exists a splitting exact sequence (obtained from the locally trivial fibration $  F (n) \rightarrow F (n - 1) $
 +
with fibre $  \mathbf R  ^ {2} \setminus  (a _ {1} \dots a _ {n - 1 }  ) $):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470122.png" /></td> </tr></table>
+
$$
 +
1  \rightarrow  F _ {n - 1 }  \rightarrow  K (n)  \rightarrow \
 +
K (n - 1)  \rightarrow  1,
 +
$$
  
 
which leads to the normal series
 
which leads to the normal series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470123.png" /></td> </tr></table>
+
$$
 +
K (n)  = \
 +
A _ {n}  \supset \dots \supset \
 +
A _ {1}  \supset  A _ {0}  = \
 +
F _ {n - 1 }
 +
$$
  
with free factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470124.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470125.png" /> has a "component"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470126.png" /> isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470127.png" />. Each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470128.png" /> can be expressed uniquely in the form
+
with free factors $  A _ {i} / A _ {i - 1 }  $,  
 +
where $  A _ {i} $
 +
has a "component"  $  U _ {n - i }  $
 +
isomorphic to $  K (n - i - 1) $.  
 +
Each element $  \omega \in B (n) $
 +
can be expressed uniquely in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470129.png" /></td> </tr></table>
+
$$
 +
\omega  = \
 +
\omega _ {2} \dots \omega _ {n} \pi _  \omega  ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470130.png" /> is a selected representative for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470131.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470132.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470133.png" />. The reduction of a braid to this form is known as its dressing. This solves the word (identity) problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470135.png" />.
+
where $  \pi _  \omega  $
 +
is a selected representative for $  S  ^  \omega  $
 +
in $  B (n) $
 +
and $  \omega _ {i} \in A _ {n - i + 1 }  \cap U _ {i} $.  
 +
The reduction of a braid to this form is known as its dressing. This solves the word (identity) problem in $  B (n) $.
  
A presentation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470136.png" /> is as follows: generators (see Fig. e):
+
A presentation of $  K (n) $
 +
is as follows: generators (see Fig. e):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470137.png" /></td> </tr></table>
+
$$
 +
A _ {ij}  = \
 +
\sigma _ {j - 1 }  \sigma _ {j} \sigma _ {j + 1 }  \sigma _ {i}  ^ {2}
 +
\sigma _ {j + 1 }  \sigma _ {j} \sigma _ {j - 1 }  \in  A _ {i} \cap U _ {n - i + 1 }  ;
 +
$$
  
 
relations:
 
relations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470138.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\left .
  
This presentation may be obtained as a presentation of the kernel of the natural homomorphism into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470139.png" /> of the abstract group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470140.png" /> defined by the presentation (1) with the aid of the [[Schreier system|Schreier system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470142.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470143.png" />.
+
\begin{array}{ll}
 +
A _ {rs} A _ {ij}  =  A _ {ij} A _ {rs}  &\textrm{ if }  r \leq  s < i < j  \\
 +
{}  &\textrm{ or }  i < r < s < j;  \\
 +
A _ {rj} A _ {ij}  = A _ {rj}  &\textrm{ if }  i = s;  \\
 +
A _ {ij} A _ {sj} A _ {ij} A _ {ij} = A _ {sj}  &\textrm{ if }  r = i < j < s;  \\
 +
A _ {rj} A _ {sj} A _ {rj}  ^ {-1}
 +
A _ {sj}  ^ {-1} A _ {ij} A _ {rj} A _ {sj}  = \
 +
A _ {sj} A _ {rj}  &\textrm{ if }  r < i < s < j. \\
 +
\end{array}
  
The centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470144.png" /> is the infinite cyclic group generated by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470145.png" />. The commutator group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470146.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470147.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470148.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470149.png" /> is isomorphic to the free group of rank 2, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470150.png" /> is isomorphic to semi-direct product of two such groups. The quotient group modulo the commutator subgroup is an infinite cyclic group, generated by the images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470151.png" />. There are no elements of finite order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470152.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470153.png" /> is mapped onto itself by endomorphisms with non-Abelian image. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470154.png" /> is a fully-characteristic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470155.png" />, and also of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470156.png" /> (see [[#References|[15]]]).
+
\right \}
 +
$$
  
Solving the conjugacy problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470157.png" /> is much more complicated than solving the word problem. There is a unique Garside normal form of a braid, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470158.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470159.png" /> is what is known as a Garside element and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470160.png" /> is a positive braid, i.e. a braid the representation of which in terms of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470161.png" /> has positive indices. With any braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470162.png" /> one can associate, using finitely many operations depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470163.png" /> (conjugation with certain elements, choice of elements of maximum degree, etc.), a certain set of words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470164.png" />, from which one selects a word in normal form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470165.png" /> with minimal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470166.png" />. This is a so-called upper form of the braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470167.png" />. It turns out that two braids are conjugate if and only if they have a same upper form (see [[#References|[7]]]).
+
This presentation may be obtained as a presentation of the kernel of the natural homomorphism into  $  S (n) $
 +
of the abstract group  $  B (n) $
 +
defined by the presentation (1) with the aid of the [[Schreier system|Schreier system]] $  \prod _ {j = 2 }  ^ {n} M _ {j} k _ {j} $,
 +
$  j \geq  k _ {j} \geq  n $,
 +
where  $  M _ {ji} = \sigma _ {j - 1 }  \dots \sigma _ {i} $.
  
The Burau representation of the braid group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470168.png" /> in the group of matrices over the ring of polynomials in one variable with integer coefficients is defined by the correspondence
+
The centre of $  B (n) $
 +
is the infinite cyclic group generated by the element  $  ( \sigma _ {1} \dots \sigma _ {n} )  ^ {n} $.  
 +
The commutator group  $  B ^ { \prime } (n) $
 +
coincides with  $  B ^ { \prime\prime } (n) $
 +
for  $  n \geq  5 $;
 +
$  B ^ { \prime } (3) $
 +
is isomorphic to the free group of rank 2, and  $  B ^ { \prime } (4) $
 +
is isomorphic to semi-direct product of two such groups. The quotient group modulo the commutator subgroup is an infinite cyclic group, generated by the images of $  \sigma _ {i} $.
 +
There are no elements of finite order in $  B (n) $.
 +
The group  $  K (n) $
 +
is mapped onto itself by endomorphisms with non-Abelian image. In particular,  $  K (n) \cap B ^ { \prime } (n) $
 +
is a fully-characteristic subgroup of  $  B (n) $,
 +
and also of  $  K (n) $(
 +
see [[#References|[15]]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470169.png" /></td> </tr></table>
+
Solving the conjugacy problem in  $  B (n) $
 +
is much more complicated than solving the word problem. There is a unique Garside normal form of a braid,  $  \omega = \Delta  ^ {m} \Omega $,
 +
where  $  \Delta = ( \sigma _ {1} {} \dots \sigma _ {n - 1 }  ) ( \sigma _ {1} \dots \sigma _ {n - 2 }  ) \dots ( \sigma _ {1} \sigma _ {2} ) \sigma _ {1} $
 +
is what is known as a Garside element and  $  \Omega $
 +
is a positive braid, i.e. a braid the representation of which in terms of the elements  $  \sigma _ {i} $
 +
has positive indices. With any braid  $  \omega $
 +
one can associate, using finitely many operations depending on  $  i $(
 +
conjugation with certain elements, choice of elements of maximum degree, etc.), a certain set of words  $  \Sigma ( \omega ) $,
 +
from which one selects a word in normal form  $  \Delta  ^ {+} T $
 +
with minimal  $  T $.  
 +
This is a so-called upper form of the braid  $  \omega $.  
 +
It turns out that two braids are conjugate if and only if they have a same upper form (see [[#References|[7]]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470170.png" /> is the identity matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470171.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470172.png" /> is the reduced Alexander matrix (see [[Alexander invariants|Alexander invariants]]) of the link obtained by closing the braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470173.png" /> (see below). For a pure braid one obtains the full Alexander matrix from the analogous Gassner matrix. The problem of whether these representations are faithful is still (1982) unsolved (see [[#References|[2]]]).
+
The Burau representation of the braid group  $  B (n) $
 +
in the group of matrices over the ring of polynomials in one variable with integer coefficients is defined by the correspondence
  
The fact that the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470174.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470175.png" /> are aspherical makes it possible to evaluate the homology of braid groups.
+
$$
 +
b ( \omega ): \sigma _ {i}  \rightarrow \
 +
\left \|
  
The homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470177.png" /> (see [[#References|[16]]]). Homologically, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470178.png" /> coincides with the product of unions of circles in which the number of circles runs from one through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470179.png" />. The homology ring is isomorphic to the exterior graded ring generated by the one-dimensional elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470181.png" />, with relations
+
\begin{array}{cccc}
 +
I _ {i - 1 }  & 0  & 0 & 0 \\
 +
0 &i - t  & t  & 0 \\
 +
0  & 1  & 0  & 0  \\
 +
0  & 0  & 0 &I _ {n - k - 1 }  \\
 +
\end{array}
 +
\
 +
\right \| ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470182.png" /></td> </tr></table>
+
where  $  I _ {k} $
 +
is the identity matrix of order  $  k $.
 +
The matrix  $  b ( \omega ) - I _ {n} $
 +
is the reduced Alexander matrix (see [[Alexander invariants|Alexander invariants]]) of the link obtained by closing the braid  $  \omega $(
 +
see below). For a pure braid one obtains the full Alexander matrix from the analogous Gassner matrix. The problem of whether these representations are faithful is still (1982) unsolved (see [[#References|[2]]]).
  
As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470183.png" /> one can take the forms
+
The fact that the spaces  $  F (n) $
 +
and  $  G (n) $
 +
are aspherical makes it possible to evaluate the homology of braid groups.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470184.png" /></td> </tr></table>
+
The homology of  $  K (n) $(
 +
see [[#References|[16]]]). Homologically,  $  K (n) $
 +
coincides with the product of unions of circles in which the number of circles runs from one through  $  n - 1 $.
 +
The homology ring is isomorphic to the exterior graded ring generated by the one-dimensional elements  $  \omega _ {ij} = \omega _ {ji} $,
 +
$  1 \leq  i \leq  j \leq  n $,
 +
with relations
  
corresponding to passage along the diagonals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470185.png" />.
+
$$
 +
\omega _ {kl} \omega _ {lm} +
 +
\omega _ {lm} \omega _ {mk} +
 +
\omega _ {mk} \omega _ {kl}  = 0.
 +
$$
  
The homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470187.png" /> (see [[#References|[8]]], [[#References|[12]]]). The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470188.png" /> can be extended to an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470189.png" />; the induced homomorphism of cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470190.png" /> is epimorphic, i.e. the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470191.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470192.png" /> generated by the Stiefel–Whitney classes.
+
As  $  \omega _ {kl} $
 +
one can take the forms
  
There is a natural mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470193.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470194.png" />, the two-fold loop space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470195.png" />, i.e. the space of spheroids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470196.png" /> (choose small discs about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470197.png" /> points, then map these discs canonically with degree 1 into a sphere, mapping the entire complement onto a point). This mapping (see [[#References|[14]]]) establishes a homology equivalence of the limit space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470198.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470199.png" /> (the subscript indicates that one chooses the component of spheroids of degree 0). As to the unstable homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470200.png" />, it has been proved [[#References|[16]]] that they are finite, stabilize at height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470201.png" /> and satisfy the recurrence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470202.png" />. There is a description of the calculation of these groups [[#References|[17]]].
+
$$
 +
 
 +
\frac{1}{2 \pi i }
 +
 
 +
\frac{dz _ {k} - dz _ {l} }{z _ {k} - z _ {l} }
 +
,
 +
$$
 +
 
 +
corresponding to passage along the diagonals  $  z _ {k} = z _ {l} $.
 +
 
 +
The homology of  $  B (n) $(
 +
see [[#References|[8]]], [[#References|[12]]]). The homomorphism  $  B (n) \rightarrow S (n) $
 +
can be extended to an imbedding  $  S (n) \rightarrow O (n) $;
 +
the induced homomorphism of cohomology spaces  $  H  ^ {*} (O (n)) \rightarrow H  ^ {*} (B (n)) $
 +
is epimorphic, i.e. the cohomology spaces  $  \mathop{\rm mod}  2 $
 +
of the group  $  B (n) $
 +
generated by the Stiefel–Whitney classes.
 +
 
 +
There is a natural mapping of $  G (n) $
 +
into $  \Omega  ^ {2} S  ^ {2} $,  
 +
the two-fold loop space of $  S  ^ {2} $,  
 +
i.e. the space of spheroids $  S  ^ {2} \rightarrow S  ^ {2} $(
 +
choose small discs about $  n $
 +
points, then map these discs canonically with degree 1 into a sphere, mapping the entire complement onto a point). This mapping (see [[#References|[14]]]) establishes a homology equivalence of the limit space $  G ( \infty ) $
 +
and $  ( \Omega  ^ {2} S  ^ {2} ) _ {0} $(
 +
the subscript indicates that one chooses the component of spheroids of degree 0). As to the unstable homology groups of $  B (n) $,  
 +
it has been proved [[#References|[16]]] that they are finite, stabilize at height $  n $
 +
and satisfy the recurrence relation $  H  ^ {i} B (2n + 1) = H  ^ {i} B (2n) $.  
 +
There is a description of the calculation of these groups [[#References|[17]]].
  
 
==Applications and generalizations.==
 
==Applications and generalizations.==
  
 
+
1) A closed braid is a link (an $  n $-
1) A closed braid is a link (an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470203.png" />-component knot) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470204.png" /> each component of which transversally cuts out half-planes bounded by the same straight line: the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470205.png" /> of the closed braid (see Fig. f).
+
component knot) in $  \mathbf R  ^ {3} $
 +
each component of which transversally cuts out half-planes bounded by the same straight line: the axis $  l $
 +
of the closed braid (see Fig. f).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017470f.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b017470f.gif" />
Line 132: Line 420:
 
Figure: b017470g
 
Figure: b017470g
  
A braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470206.png" /> generates a closed braid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470207.png" /> (the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470208.png" />) in the following way. Consider a cylinder with bases on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470210.png" />, the interior of which contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470211.png" />. Let this cylinder be deformed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470212.png" /> so that its elements become circles with centres on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470213.png" />, its bases coincide and each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470214.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470215.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470216.png" /> is the union of the strings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470217.png" />. Conversely, every link in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470218.png" /> can be represented by a closed braid. To equivalent braids correspond isotopic links and, moreover, conjugate braids yield isotopic links. The converse is false, since a link may be represented by braids with different numbers of strings. In addition, the braids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470219.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470220.png" /> are not conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470221.png" />, but they correspond to isotopic links. If two closed braids are equivalent as links, they can be derived from one another by a chain of elementary transformations of two types (see Fig. g). These operations are interpreted in terms of presentations of the link group, thus yielding a reformulation of the isotopy problem for links as a question concerning the system of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470222.png" />. A presentation of the link group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470223.png" /> has the form
+
A braid $  \omega $
 +
generates a closed braid $  \widetilde \omega  $(
 +
the closure of $  \omega $)  
 +
in the following way. Consider a cylinder with bases on $  P _ {0} $
 +
and $  P _ {1} $,  
 +
the interior of which contains $  \omega $.  
 +
Let this cylinder be deformed in $  \mathbf R  ^ {3} $
 +
so that its elements become circles with centres on $  l $,  
 +
its bases coincide and each point $  a _ {i} $
 +
coincides with b _ {i} $.  
 +
Then $  \widetilde \omega  $
 +
is the union of the strings $  l _ {i} $.  
 +
Conversely, every link in $  \mathbf R  ^ {3} $
 +
can be represented by a closed braid. To equivalent braids correspond isotopic links and, moreover, conjugate braids yield isotopic links. The converse is false, since a link may be represented by braids with different numbers of strings. In addition, the braids $  \omega \sigma _ {n - 1 }  $
 +
and $  \omega \sigma _ {n - 1 }  ^ {-1} $
 +
are not conjugate in $  B (n) $,  
 +
but they correspond to isotopic links. If two closed braids are equivalent as links, they can be derived from one another by a chain of elementary transformations of two types (see Fig. g). These operations are interpreted in terms of presentations of the link group, thus yielding a reformulation of the isotopy problem for links as a question concerning the system of groups $  B (n) $.  
 +
A presentation of the link group of $  \widetilde \omega  $
 +
has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470224.png" /></td> </tr></table>
+
$$
 +
\{ {y _ {1} \dots y _ {n} } : {
 +
y _ {i} =
 +
A _ {i} y _ {k _ {i}  } A _ {i}  ^ {-1} } \}
 +
,
 +
$$
  
where the relations are defined by braid automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470225.png" />. Conversely, any such relation defines a braid.
+
where the relations are defined by braid automorphisms b ^  \omega  $.  
 +
Conversely, any such relation defines a braid.
  
2) If one cuts a surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470226.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470227.png" /> non-intersecting cuts so as to obtain a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470228.png" /> holes, then the homeomorphisms of this sphere with holes that fix points on the edges of the holes define homeomorphisms of the surface which fix the cuts and are themselves defined up to isotopy by the elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470229.png" />. This yields a representation of the braid group in the homotopy group of the surface. Similarly one constructs a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470230.png" />. These representations are used in studying Heegaard diagrams of three-dimensional manifolds (cf. [[Three-dimensional manifold|Three-dimensional manifold]]).
+
2) If one cuts a surface of genus $  g $
 +
with $  g $
 +
non-intersecting cuts so as to obtain a sphere with $  2g $
 +
holes, then the homeomorphisms of this sphere with holes that fix points on the edges of the holes define homeomorphisms of the surface which fix the cuts and are themselves defined up to isotopy by the elements of the group $  K (2g) $.  
 +
This yields a representation of the braid group in the homotopy group of the surface. Similarly one constructs a representation of $  B (2g) $.  
 +
These representations are used in studying Heegaard diagrams of three-dimensional manifolds (cf. [[Three-dimensional manifold|Three-dimensional manifold]]).
  
3) By identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470231.png" /> with the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470232.png" /> and associating with any unordered set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470233.png" /> points in the plane a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470234.png" /> having these points as roots, one can identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470235.png" /> with the space of polynomials with non-zero discriminants. This has made it possible to obtain several results concerning the non-representability of algebraic functions by superpositions of functions in fewer variables (see [[#References|[16]]]).
+
3) By identifying $  \mathbf R  ^ {2} $
 +
with the complex plane $  \mathbf C  ^ {1} $
 +
and associating with any unordered set of $  n $
 +
points in the plane a polynomial of degree $  n $
 +
having these points as roots, one can identify $  G (n) $
 +
with the space of polynomials with non-zero discriminants. This has made it possible to obtain several results concerning the non-representability of algebraic functions by superpositions of functions in fewer variables (see [[#References|[16]]]).
  
4) Configuration spaces for arbitrary spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470236.png" /> are defined in analogy with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470237.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470238.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470239.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470240.png" />. The fundamental groups of these spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470241.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470242.png" />, are called the braid group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470243.png" /> and the pure braid group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470244.png" />, respectively. For a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470245.png" /> of dimension exceeding 2, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470246.png" />, and this group is of no interest. For two-dimensional manifolds, one has a natural imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470247.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470248.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470249.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470250.png" /> induced by an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470251.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470252.png" /> is neither a sphere nor a projective space, one obtains an exact sequence
+
4) Configuration spaces for arbitrary spaces $  X $
 +
are defined in analogy with $  G (n) $
 +
and $  F (n) $,  
 +
with $  \mathbf R  ^ {2} $
 +
replaced by $  X $.  
 +
The fundamental groups of these spaces, $  B (X) $
 +
and $  K (X) $,  
 +
are called the braid group of the space $  X $
 +
and the pure braid group of the space $  X $,  
 +
respectively. For a manifold $  M  ^ {n} $
 +
of dimension exceeding 2, $  \pi _ {1} F _ {n} (X) \approx \prod _ {i = 1 }  ^ {n} \pi _ {i} M (i) $,  
 +
and this group is of no interest. For two-dimensional manifolds, one has a natural imbedding of $  B (n) $
 +
and $  K (n) $
 +
into $  B _ {n} (M  ^ {2} ) $
 +
and $  K _ {n} (M  ^ {2} ) $
 +
induced by an imbedding $  \mathbf R  ^ {2} \subset  M  ^ {2} $.  
 +
If $  M  ^ {2} $
 +
is neither a sphere nor a projective space, one obtains an exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470253.png" /></td> </tr></table>
+
$$
 +
1  \rightarrow  \pi _ {1} K (2)
 +
\rightarrow ^ { e }  \
 +
\pi _ {1} K _ {n} (M  ^ {2} )
 +
\rightarrow  \prod _ {i = 1 } ^ { n }
 +
\pi _ {i} M _ {(i)}  ^ {2} ;
 +
$$
  
for the sphere, the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470254.png" /> is an epimorphism, obtained by adding to (1) the single relation
+
for the sphere, the homomorphism $  e $
 +
is an epimorphism, obtained by adding to (1) the single relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470255.png" /></td> </tr></table>
+
$$
 +
\sigma _ {s} \dots
 +
\sigma _ {n - 2 }
 +
\sigma _ {n - 1 }  ^ {2}
 +
\sigma _ {n - 2 }  \dots
 +
\sigma _ {1}  = 1.
 +
$$
  
5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470256.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470257.png" />-sheeted covering, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470258.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470259.png" /> is a loop in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470260.png" />, is a loop in the configuration space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470261.png" />, and this defines a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470262.png" /> which strengthens the monodromy of the covering and has applications in algebraic geometry.
+
5) If $  p: X \rightarrow Y $
 +
is a $  k $-
 +
sheeted covering, then $  p  ^ {-1} \alpha $,  
 +
where $  \alpha $
 +
is a loop in $  Y $,  
 +
is a loop in the configuration space $  X $,  
 +
and this defines a homomorphism $  \pi _ {1} Y \rightarrow B _ {k} (X) $
 +
which strengthens the monodromy of the covering and has applications in algebraic geometry.
  
6) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470263.png" /> be the complexification of a real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470264.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470265.png" /> be a finite irreducible group generated by reflections acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470266.png" /> (hence also in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470267.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470268.png" /> be generating reflections in the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470269.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470270.png" /> be their union. Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470271.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470272.png" /> be the quotient space. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470273.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470274.png" /> are called Brieskorn groups and constitute natural generalizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470275.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470276.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470277.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470278.png" /> has a presentation of the form
+
6) Let $  V ^ {\mathbf C } $
 +
be the complexification of a real vector space $  V $
 +
and let $  W $
 +
be a finite irreducible group generated by reflections acting in $  V $(
 +
hence also in $  V ^ {\mathbf C } $).  
 +
Let $  s _ {i} $
 +
be generating reflections in the planes $  P _ {i} \subset  V $
 +
and let $  D $
 +
be their union. Finally, let $  V ^ {\mathbf C } /D = Y _ {W} $
 +
and let $  X _ {W} $
 +
be the quotient space. The groups $  \pi _ {1} Y _ {W} $
 +
and $  \pi _ {1} X _ {W} $
 +
are called Brieskorn groups and constitute natural generalizations of $  K (n) $
 +
and $  B (n) $.  
 +
If $  \mathop{\rm ord} (s _ {i} s _ {j} ) = m _ {ij} $,  
 +
then $  \pi _ {1} X _ {W} $
 +
has a presentation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470279.png" /></td> </tr></table>
+
$$
 +
\sigma _ {i} \sigma _ {j} \sigma _ {i} \dots  = \
 +
\sigma _ {j} \sigma _ {i} \sigma _ {j} \dots ,
 +
$$
  
where the number of factors on each side is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470280.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470281.png" /> here corresponds to a Weyl chamber). It has been proved for these groups that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470282.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470283.png" /> are spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470284.png" />, and the conjugacy problem has been solved. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017470/b017470285.png" /> appear in algebraic geometry as complements to the discriminant of versal deformations of rational singularities (see [[#References|[12]]], [[#References|[13]]]).
+
where the number of factors on each side is equal to $  m _ {ij} $(
 +
$  \sigma _ {i} $
 +
here corresponds to a Weyl chamber). It has been proved for these groups that $  X _ {W} $
 +
and $  Y _ {W} $
 +
are spaces of type $  K ( \pi , l) $,  
 +
and the conjugacy problem has been solved. The spaces $  X _ {W} $
 +
appear in algebraic geometry as complements to the discriminant of versal deformations of rational singularities (see [[#References|[12]]], [[#References|[13]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Artin,   "Theory of braids" ''Ann. of Math.'' , '''48''' (1947) pp. 643–649</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.S. Birman,   "Braids, links and mapping class groups" , Princeton Univ. Press (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Burau,   "Ueber Zopfinvarianten" ''Abh. Math. Sem. Univ. Hamburg'' , '''9''' (1932) pp. 117–124</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.A. Markov,   "Foundations of the algebraic theory of braids" ''Trudy Mat. Inst. Steklov.'' , '''16''' (1945) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> B. Gassner,   "On braid groups" ''Abh. Math. Sem. Univ. Hamburg'' , '''25''' (1961) pp. 10–22</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E. Fadell,   L. Neuwirth,   "Configuration spaces" ''Math. Scand.'' , '''10''' (1962) pp. 111–118</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> F.A. Garside,   "The braid group and other groups" ''Quart. J. Math.'' , '''20''' : 4 (1969) pp. 235–254</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> D.B. Fuks,   "Cohomology of braid groups mod 2" ''Funktional. Anal. i Prilozhen.'' , '''4''' : 2 (1970) pp. 62–73 (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> V.I. Arnol'd,   "On cohomology classes of algebraic functions that are preserved under Tschirnhausen transformations" ''Funktional. Anal. i Prilozhen.'' , '''1''' (1970) pp. 84–85 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> E.A. Gorin,   V.Ya. Lin,   "Algebraic equations with continuous coefficients and some problems in the theory of braids" ''Mat. Sb.'' , '''78''' (1969) pp. 579–610 (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.I. Arnol'd,   "On certain topological invariants of algebraic functions" ''Trudy Moskov. Mat. Obshch.'' , '''21''' (1970) pp. 27–46 (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> E. Brieskorn,   ''Matematika'' , '''18''' : 3 (1974) pp. 46–59</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> E. Brieskorn,   K. Saito,   "Artin Gruppen und Coxeter Gruppen" ''Invent. Math.'' , '''17''' (1972) pp. 245–271</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> P. Deligne,   "Les immeubles des groupes de tresses généralisés" ''Invent. Math.'' , '''17''' : 4 (1972) pp. 273–302</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> V.Ya. Lin,   "Representations of permutation braids" ''Uspekhi Mat. Nauk.'' , '''29''' : 1 (1974) pp. 173–174 (In Russian)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> V.I. Arnol'd,   "The ring of cohomology groups of crossed braids" ''Mat. Zametki'' , '''5''' : 2 (1969) pp. 227–231 (In Russian)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> V.Ya. Lin,   "Artin braids and related groups and spaces" ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''17''' (1979) pp. 159–227 (In Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Artin, "Theory of braids" ''Ann. of Math.'' , '''48''' (1947) pp. 643–649 {{MR|0019087}} {{ZBL|0030.17703}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.S. Birman, "Braids, links and mapping class groups" , Princeton Univ. Press (1974) {{MR|0375281}} {{ZBL|0305.57013}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Burau, "Ueber Zopfinvarianten" ''Abh. Math. Sem. Univ. Hamburg'' , '''9''' (1932) pp. 117–124 {{MR|}} {{ZBL|0006.03401}} {{ZBL|58.0614.03}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.A. Markov, "Foundations of the algebraic theory of braids" ''Trudy Mat. Inst. Steklov.'' , '''16''' (1945) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> B. Gassner, "On braid groups" ''Abh. Math. Sem. Univ. Hamburg'' , '''25''' (1961) pp. 10–22 {{MR|0130309}} {{ZBL|0111.03002}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E. Fadell, L. Neuwirth, "Configuration spaces" ''Math. Scand.'' , '''10''' (1962) pp. 111–118 {{MR|0141127}} {{MR|0141126}} {{ZBL|0136.44104}} {{ZBL|0122.17803}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> F.A. Garside, "The braid group and other groups" ''Quart. J. Math.'' , '''20''' : 4 (1969) pp. 235–254 {{MR|0248801}} {{ZBL|0194.03303}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> D.B. Fuks, "Cohomology of braid groups mod 2" ''Funktional. Anal. i Prilozhen.'' , '''4''' : 2 (1970) pp. 62–73 (In Russian) {{MR|274463}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> V.I. Arnol'd, "On cohomology classes of algebraic functions that are preserved under Tschirnhausen transformations" ''Funktional. Anal. i Prilozhen.'' , '''1''' (1970) pp. 84–85 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> E.A. Gorin, V.Ya. Lin, "Algebraic equations with continuous coefficients and some problems in the theory of braids" ''Mat. Sb.'' , '''78''' (1969) pp. 579–610 (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.I. Arnol'd, "On certain topological invariants of algebraic functions" ''Trudy Moskov. Mat. Obshch.'' , '''21''' (1970) pp. 27–46 (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> E. Brieskorn, ''Matematika'' , '''18''' : 3 (1974) pp. 46–59 {{MR|0360085}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> E. Brieskorn, K. Saito, "Artin Gruppen und Coxeter Gruppen" ''Invent. Math.'' , '''17''' (1972) pp. 245–271 {{MR|0323910}} {{ZBL|0243.20037}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> P. Deligne, "Les immeubles des groupes de tresses généralisés" ''Invent. Math.'' , '''17''' : 4 (1972) pp. 273–302 {{MR|0422673}} {{ZBL|0238.20034}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> V.Ya. Lin, "Representations of permutation braids" ''Uspekhi Mat. Nauk.'' , '''29''' : 1 (1974) pp. 173–174 (In Russian)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> V.I. Arnol'd, "The ring of cohomology groups of crossed braids" ''Mat. Zametki'' , '''5''' : 2 (1969) pp. 227–231 (In Russian)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> V.Ya. Lin, "Artin braids and related groups and spaces" ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''17''' (1979) pp. 159–227 (In Russian) {{MR|0584570}} {{ZBL|}} </TD></TR></table>
 +
 
 +
{{OldImage}}

Latest revision as of 08:41, 26 March 2023


The branch of topology and algebra concerned with braids, the groups formed by their equivalence classes and various generalizations of these groups [1].

A braid on $ n $ strings is an object consisting of two parallel planes $ P _ {0} $ and $ P _ {1} $ in three-dimensional space $ \mathbf R ^ {3} $, containing two ordered sets of points $ a _ {1} \dots a _ {n} \in P _ {0} $ and $ b _ {1} \dots b _ {n} \in P _ {1} $, and of $ n $ simple non-intersecting arcs $ l _ {1} \dots l _ {n} $, intersecting each parallel plane $ P _ {t} $ between $ P _ {0} $ and $ P _ {1} $ exactly once and joining the points $ \{ a _ {i} \} $ to $ \{ b _ {i} \} $, $ i = 1 \dots n $. It is assumed that the $ a _ {i} $' s lie on a straight line $ L _ {a} $ in $ P _ {0} $ and the $ b _ {i} $' s on a straight line $ L _ {b} $ in $ P _ {1} $ parallel to $ L _ {a} $; moreover, $ b _ {i} $ lies beneath $ a _ {i} $ for each $ i $( see Fig. a). Braids can be represented in the projection on the plane passing through $ L _ {a} $ and $ L _ {b} $; this projection can be brought into general position in such a way that there are only finitely many double points, each two of which lie at different levels, and the intersections are transversal.

Figure: b017470a

Figure: b017470b

The string $ l _ {i} $ of a braid $ \omega $ joins $ a _ {i} $ to $ b _ {k _ {i} } $ and so defines a permutation

$$ S ^ \omega = \ \left ( \ \begin{array}{ccc} 1 &\dots & n \\ k _ {1} &\dots &k _ {n} \\ \end{array} \ \right ) . $$

If this is the identity permutation, $ \omega $ is called a coloured (or pure) braid. The transposition $ (i i + 1) $ corresponds to a simple braid $ \sigma _ {i} $( see Fig. b).

On the set of all braids on $ n $ strings with fixed $ P _ {0} $, $ P _ {1} $, $ \{ a _ {i} \} $, $ \{ b _ {i} \} $, one introduces the equivalence relation defined by homeomorphisms $ h: \Pi \rightarrow \Pi $, where $ \Pi $ is the region between $ P _ {0} $ and $ P _ {1} $, which reduce to the identity on $ P _ {0} \cup P _ {1} $; it may be assumed that $ h (P _ {t} ) = P _ {t} $. Braids $ \alpha $ and $ \beta $ are equivalent if there exists a homeomorphism with the above properties such that $ h ( \alpha ) = \beta $.

The equivalence classes — which are still called braids — form the braid group $ B (n) $ with respect to the operation defined as follows. Place a copy $ \Pi ^ \prime $ of the domain $ \Pi $ above another copy $ \Pi ^ {\prime\prime} $, in such a way that $ P _ {0} ^ {\prime\prime} $ coincides with $ P _ {1} ^ \prime $, $ a _ {i} ^ {\prime\prime} $ with $ b _ {i} ^ \prime $, and then compress $ \Pi ^ \prime \cup \Pi ^ {\prime\prime} $ to half its "height" . The images of the braids $ \omega ^ \prime \in \Pi ^ \prime $ and $ \omega ^ {\prime\prime} \in \Pi ^ {\prime\prime} $ produce a braid $ \omega ^ \prime \omega ^ {\prime\prime} $, with string $ l _ {i} $ obtained by extending $ l _ {i} ^ \prime $ with $ l _ {k _ {i} } ^ {\prime\prime} $, where $ k _ {i} \in S ^ {\omega ^ \prime (i) } $. The identity braid is the equivalence class containing the braid with $ n $ parallel segments; the inverse $ \omega ^ {-1} $ of a braid $ \omega $ is defined by reflection in the plane $ P _ {1/2} $. For the condition $ \omega \omega ^ {-1} = \epsilon $ see Fig. c.

Figure: b017470c

The mapping $ \omega \rightarrow S ^ \omega $ defines an epimorphism of $ B (n) $ onto the group $ S (n) $ of permutations of $ n $ elements, the kernel of this epimorphism is the subgroup $ K (n) $ of all pure braids, so that one has an exact sequence

$$ 1 \rightarrow K (n) \rightarrow B (n) \rightarrow S (n) \rightarrow 1. $$

The braid group $ B (n) $ has two principal interpretations. The first, as a configuration space, is obtained by identifying the planes $ P _ {t} $ via vertical projection onto $ P _ {0} $, under which the images of the points $ a _ {it } = l _ {i} \cap P _ {t} $, considered as $ t $ varies from 0 to 1, form the trace of an isotopy $ \phi _ {t} ^ \omega $ of the set $ \cup a _ {i} $ along $ P _ {0} $; one has $ \phi _ {1} ^ \omega ( \cup a _ {i} ) = \cup a _ {i} $. Consider the space of unordered sequences $ G (n) $ of $ n $ pairwise distinct points of the plane; then each braid corresponds in one-to-one fashion to a class of homotopy loops in this space, and one has an isomorphism

$$ \beta : B (n) \rightarrow \ \pi _ {1} G (n). $$

For pure braids one has an analogously constructed isomorphism

$$ \alpha : K (n) \rightarrow \ \pi _ {1} F (n), $$

where $ F (n) $ is the space of ordered sequences of $ n $ distinct points of the plane, so that $ K (n) $ can be identified with the subgroup corresponding to the covering

$$ p: F (n) \rightarrow G (n) = \ F (n) / S (n). $$

The second interpretation, as a homeotopy group, is obtained by extending the isotopy $ \phi _ {t} ^ \omega $ to an isotopy $ {\widetilde \phi } {} _ {t} ^ \omega $ of the plane $ P _ {0} $ that coincides with the identity outside some disc, and such that $ {\widetilde \phi } {} _ {0} ^ \omega = \mathop{\rm id} $. For each $ t $, two such extensions differ by a homeomorphism which is the identity at the points $ a _ {it} $. A braid uniquely determines a component of the space of homeomorphisms $ Y (n) $ of the plane which map the set $ \cup a _ {i} $ onto itself, and one has an isomorphism

$$ \gamma : B (n) \rightarrow \ \pi _ {0} Y (n). $$

To each homeomorphism $ h \in Y $ corresponds an automorphism of the free group of rank $ n: $ $ F _ {n} = \pi _ {1} ( \mathbf R ^ {2} \setminus \cup a _ {i} ) $, defined up to an inner automorphism, which in turn yields a homomorphism $ B (n) \rightarrow \mathop{\rm Out} F _ {n} = \mathop{\rm Aut} F _ {n} / \mathop{\rm Inn} F _ {n} $. The elements of the image are called braid automorphisms of the free group. In particular, corresponding to the braid $ \sigma _ {i} $ one has an automorphism

$$ \overline \sigma \; _ {i} :\ \overline \sigma \; _ {i} (x _ {i} ) = \ x _ {i + 1 } ,\ \ \overline \sigma \; _ {i} (x _ {i + 1 } ) = \ x _ {i + 1 } x _ {i} x _ {i + 1 } ^ {-1} ,\ \ \overline \sigma \; _ {i} (x _ {j} ) = x _ {j} , $$

if $ j \neq i, i + 1 $( $ \{ x _ {i} \} $ is a set of generators of $ F _ {n} $). Any braid automorphism $ \alpha $ possesses the following properties:

$$ \alpha (x _ {i} ) = \ A _ {i} x _ {i} A _ {i} ^ {-1} ,\ \ \alpha \left ( \prod _ {i = 1 } ^ { n } x _ {i} \right ) = \ \prod _ {i = 1 } ^ { n } x _ {i} , $$

up to an inner automorphism (for the meaning of $ A _ {i} $, see below); these properties characterize braid automorphisms.

The braids $ \sigma _ {i} , 1 \leq i \leq n - 1 $, are the generators of the group $ B (n) $, i.e. $ \omega = \sigma _ {k _ {1} } \dots \sigma _ {k _ {m} } $, with

$$ \tag{1 } \left . \begin{array}{ll} \sigma _ {i} \sigma _ {j} = \ \sigma _ {j} \sigma _ {i} & \textrm{ if } | i - j | > 1, \\ \sigma _ {i} \sigma _ {i + 1 } \sigma _ {i} = \ \sigma _ {i + 1 } \sigma _ {i} \sigma _ {i - 1 } , & 1 \leq i \leq n - 2. \\ \end{array} \ \right \} $$

It turns out that (1) is a presentation of $ B (n) $( see Fig. d).

Figure: b017470d

Figure: b017470e

There exists a splitting exact sequence (obtained from the locally trivial fibration $ F (n) \rightarrow F (n - 1) $ with fibre $ \mathbf R ^ {2} \setminus (a _ {1} \dots a _ {n - 1 } ) $):

$$ 1 \rightarrow F _ {n - 1 } \rightarrow K (n) \rightarrow \ K (n - 1) \rightarrow 1, $$

which leads to the normal series

$$ K (n) = \ A _ {n} \supset \dots \supset \ A _ {1} \supset A _ {0} = \ F _ {n - 1 } $$

with free factors $ A _ {i} / A _ {i - 1 } $, where $ A _ {i} $ has a "component" $ U _ {n - i } $ isomorphic to $ K (n - i - 1) $. Each element $ \omega \in B (n) $ can be expressed uniquely in the form

$$ \omega = \ \omega _ {2} \dots \omega _ {n} \pi _ \omega , $$

where $ \pi _ \omega $ is a selected representative for $ S ^ \omega $ in $ B (n) $ and $ \omega _ {i} \in A _ {n - i + 1 } \cap U _ {i} $. The reduction of a braid to this form is known as its dressing. This solves the word (identity) problem in $ B (n) $.

A presentation of $ K (n) $ is as follows: generators (see Fig. e):

$$ A _ {ij} = \ \sigma _ {j - 1 } \sigma _ {j} \sigma _ {j + 1 } \sigma _ {i} ^ {2} \sigma _ {j + 1 } \sigma _ {j} \sigma _ {j - 1 } \in A _ {i} \cap U _ {n - i + 1 } ; $$

relations:

$$ \tag{2 } \left . \begin{array}{ll} A _ {rs} A _ {ij} = A _ {ij} A _ {rs} &\textrm{ if } r \leq s < i < j \\ {} &\textrm{ or } i < r < s < j; \\ A _ {rj} A _ {ij} = A _ {rj} &\textrm{ if } i = s; \\ A _ {ij} A _ {sj} A _ {ij} A _ {ij} = A _ {sj} &\textrm{ if } r = i < j < s; \\ A _ {rj} A _ {sj} A _ {rj} ^ {-1} A _ {sj} ^ {-1} A _ {ij} A _ {rj} A _ {sj} = \ A _ {sj} A _ {rj} &\textrm{ if } r < i < s < j. \\ \end{array} \right \} $$

This presentation may be obtained as a presentation of the kernel of the natural homomorphism into $ S (n) $ of the abstract group $ B (n) $ defined by the presentation (1) with the aid of the Schreier system $ \prod _ {j = 2 } ^ {n} M _ {j} k _ {j} $, $ j \geq k _ {j} \geq n $, where $ M _ {ji} = \sigma _ {j - 1 } \dots \sigma _ {i} $.

The centre of $ B (n) $ is the infinite cyclic group generated by the element $ ( \sigma _ {1} \dots \sigma _ {n} ) ^ {n} $. The commutator group $ B ^ { \prime } (n) $ coincides with $ B ^ { \prime\prime } (n) $ for $ n \geq 5 $; $ B ^ { \prime } (3) $ is isomorphic to the free group of rank 2, and $ B ^ { \prime } (4) $ is isomorphic to semi-direct product of two such groups. The quotient group modulo the commutator subgroup is an infinite cyclic group, generated by the images of $ \sigma _ {i} $. There are no elements of finite order in $ B (n) $. The group $ K (n) $ is mapped onto itself by endomorphisms with non-Abelian image. In particular, $ K (n) \cap B ^ { \prime } (n) $ is a fully-characteristic subgroup of $ B (n) $, and also of $ K (n) $( see [15]).

Solving the conjugacy problem in $ B (n) $ is much more complicated than solving the word problem. There is a unique Garside normal form of a braid, $ \omega = \Delta ^ {m} \Omega $, where $ \Delta = ( \sigma _ {1} {} \dots \sigma _ {n - 1 } ) ( \sigma _ {1} \dots \sigma _ {n - 2 } ) \dots ( \sigma _ {1} \sigma _ {2} ) \sigma _ {1} $ is what is known as a Garside element and $ \Omega $ is a positive braid, i.e. a braid the representation of which in terms of the elements $ \sigma _ {i} $ has positive indices. With any braid $ \omega $ one can associate, using finitely many operations depending on $ i $( conjugation with certain elements, choice of elements of maximum degree, etc.), a certain set of words $ \Sigma ( \omega ) $, from which one selects a word in normal form $ \Delta ^ {+} T $ with minimal $ T $. This is a so-called upper form of the braid $ \omega $. It turns out that two braids are conjugate if and only if they have a same upper form (see [7]).

The Burau representation of the braid group $ B (n) $ in the group of matrices over the ring of polynomials in one variable with integer coefficients is defined by the correspondence

$$ b ( \omega ): \sigma _ {i} \rightarrow \ \left \| \begin{array}{cccc} I _ {i - 1 } & 0 & 0 & 0 \\ 0 &i - t & t & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 &I _ {n - k - 1 } \\ \end{array} \ \right \| , $$

where $ I _ {k} $ is the identity matrix of order $ k $. The matrix $ b ( \omega ) - I _ {n} $ is the reduced Alexander matrix (see Alexander invariants) of the link obtained by closing the braid $ \omega $( see below). For a pure braid one obtains the full Alexander matrix from the analogous Gassner matrix. The problem of whether these representations are faithful is still (1982) unsolved (see [2]).

The fact that the spaces $ F (n) $ and $ G (n) $ are aspherical makes it possible to evaluate the homology of braid groups.

The homology of $ K (n) $( see [16]). Homologically, $ K (n) $ coincides with the product of unions of circles in which the number of circles runs from one through $ n - 1 $. The homology ring is isomorphic to the exterior graded ring generated by the one-dimensional elements $ \omega _ {ij} = \omega _ {ji} $, $ 1 \leq i \leq j \leq n $, with relations

$$ \omega _ {kl} \omega _ {lm} + \omega _ {lm} \omega _ {mk} + \omega _ {mk} \omega _ {kl} = 0. $$

As $ \omega _ {kl} $ one can take the forms

$$ \frac{1}{2 \pi i } \frac{dz _ {k} - dz _ {l} }{z _ {k} - z _ {l} } , $$

corresponding to passage along the diagonals $ z _ {k} = z _ {l} $.

The homology of $ B (n) $( see [8], [12]). The homomorphism $ B (n) \rightarrow S (n) $ can be extended to an imbedding $ S (n) \rightarrow O (n) $; the induced homomorphism of cohomology spaces $ H ^ {*} (O (n)) \rightarrow H ^ {*} (B (n)) $ is epimorphic, i.e. the cohomology spaces $ \mathop{\rm mod} 2 $ of the group $ B (n) $ generated by the Stiefel–Whitney classes.

There is a natural mapping of $ G (n) $ into $ \Omega ^ {2} S ^ {2} $, the two-fold loop space of $ S ^ {2} $, i.e. the space of spheroids $ S ^ {2} \rightarrow S ^ {2} $( choose small discs about $ n $ points, then map these discs canonically with degree 1 into a sphere, mapping the entire complement onto a point). This mapping (see [14]) establishes a homology equivalence of the limit space $ G ( \infty ) $ and $ ( \Omega ^ {2} S ^ {2} ) _ {0} $( the subscript indicates that one chooses the component of spheroids of degree 0). As to the unstable homology groups of $ B (n) $, it has been proved [16] that they are finite, stabilize at height $ n $ and satisfy the recurrence relation $ H ^ {i} B (2n + 1) = H ^ {i} B (2n) $. There is a description of the calculation of these groups [17].

Applications and generalizations.

1) A closed braid is a link (an $ n $- component knot) in $ \mathbf R ^ {3} $ each component of which transversally cuts out half-planes bounded by the same straight line: the axis $ l $ of the closed braid (see Fig. f).

Figure: b017470f

Figure: b017470g

A braid $ \omega $ generates a closed braid $ \widetilde \omega $( the closure of $ \omega $) in the following way. Consider a cylinder with bases on $ P _ {0} $ and $ P _ {1} $, the interior of which contains $ \omega $. Let this cylinder be deformed in $ \mathbf R ^ {3} $ so that its elements become circles with centres on $ l $, its bases coincide and each point $ a _ {i} $ coincides with $ b _ {i} $. Then $ \widetilde \omega $ is the union of the strings $ l _ {i} $. Conversely, every link in $ \mathbf R ^ {3} $ can be represented by a closed braid. To equivalent braids correspond isotopic links and, moreover, conjugate braids yield isotopic links. The converse is false, since a link may be represented by braids with different numbers of strings. In addition, the braids $ \omega \sigma _ {n - 1 } $ and $ \omega \sigma _ {n - 1 } ^ {-1} $ are not conjugate in $ B (n) $, but they correspond to isotopic links. If two closed braids are equivalent as links, they can be derived from one another by a chain of elementary transformations of two types (see Fig. g). These operations are interpreted in terms of presentations of the link group, thus yielding a reformulation of the isotopy problem for links as a question concerning the system of groups $ B (n) $. A presentation of the link group of $ \widetilde \omega $ has the form

$$ \{ {y _ {1} \dots y _ {n} } : { y _ {i} = A _ {i} y _ {k _ {i} } A _ {i} ^ {-1} } \} , $$

where the relations are defined by braid automorphisms $ b ^ \omega $. Conversely, any such relation defines a braid.

2) If one cuts a surface of genus $ g $ with $ g $ non-intersecting cuts so as to obtain a sphere with $ 2g $ holes, then the homeomorphisms of this sphere with holes that fix points on the edges of the holes define homeomorphisms of the surface which fix the cuts and are themselves defined up to isotopy by the elements of the group $ K (2g) $. This yields a representation of the braid group in the homotopy group of the surface. Similarly one constructs a representation of $ B (2g) $. These representations are used in studying Heegaard diagrams of three-dimensional manifolds (cf. Three-dimensional manifold).

3) By identifying $ \mathbf R ^ {2} $ with the complex plane $ \mathbf C ^ {1} $ and associating with any unordered set of $ n $ points in the plane a polynomial of degree $ n $ having these points as roots, one can identify $ G (n) $ with the space of polynomials with non-zero discriminants. This has made it possible to obtain several results concerning the non-representability of algebraic functions by superpositions of functions in fewer variables (see [16]).

4) Configuration spaces for arbitrary spaces $ X $ are defined in analogy with $ G (n) $ and $ F (n) $, with $ \mathbf R ^ {2} $ replaced by $ X $. The fundamental groups of these spaces, $ B (X) $ and $ K (X) $, are called the braid group of the space $ X $ and the pure braid group of the space $ X $, respectively. For a manifold $ M ^ {n} $ of dimension exceeding 2, $ \pi _ {1} F _ {n} (X) \approx \prod _ {i = 1 } ^ {n} \pi _ {i} M (i) $, and this group is of no interest. For two-dimensional manifolds, one has a natural imbedding of $ B (n) $ and $ K (n) $ into $ B _ {n} (M ^ {2} ) $ and $ K _ {n} (M ^ {2} ) $ induced by an imbedding $ \mathbf R ^ {2} \subset M ^ {2} $. If $ M ^ {2} $ is neither a sphere nor a projective space, one obtains an exact sequence

$$ 1 \rightarrow \pi _ {1} K (2) \rightarrow ^ { e } \ \pi _ {1} K _ {n} (M ^ {2} ) \rightarrow \prod _ {i = 1 } ^ { n } \pi _ {i} M _ {(i)} ^ {2} ; $$

for the sphere, the homomorphism $ e $ is an epimorphism, obtained by adding to (1) the single relation

$$ \sigma _ {s} \dots \sigma _ {n - 2 } \sigma _ {n - 1 } ^ {2} \sigma _ {n - 2 } \dots \sigma _ {1} = 1. $$

5) If $ p: X \rightarrow Y $ is a $ k $- sheeted covering, then $ p ^ {-1} \alpha $, where $ \alpha $ is a loop in $ Y $, is a loop in the configuration space $ X $, and this defines a homomorphism $ \pi _ {1} Y \rightarrow B _ {k} (X) $ which strengthens the monodromy of the covering and has applications in algebraic geometry.

6) Let $ V ^ {\mathbf C } $ be the complexification of a real vector space $ V $ and let $ W $ be a finite irreducible group generated by reflections acting in $ V $( hence also in $ V ^ {\mathbf C } $). Let $ s _ {i} $ be generating reflections in the planes $ P _ {i} \subset V $ and let $ D $ be their union. Finally, let $ V ^ {\mathbf C } /D = Y _ {W} $ and let $ X _ {W} $ be the quotient space. The groups $ \pi _ {1} Y _ {W} $ and $ \pi _ {1} X _ {W} $ are called Brieskorn groups and constitute natural generalizations of $ K (n) $ and $ B (n) $. If $ \mathop{\rm ord} (s _ {i} s _ {j} ) = m _ {ij} $, then $ \pi _ {1} X _ {W} $ has a presentation of the form

$$ \sigma _ {i} \sigma _ {j} \sigma _ {i} \dots = \ \sigma _ {j} \sigma _ {i} \sigma _ {j} \dots , $$

where the number of factors on each side is equal to $ m _ {ij} $( $ \sigma _ {i} $ here corresponds to a Weyl chamber). It has been proved for these groups that $ X _ {W} $ and $ Y _ {W} $ are spaces of type $ K ( \pi , l) $, and the conjugacy problem has been solved. The spaces $ X _ {W} $ appear in algebraic geometry as complements to the discriminant of versal deformations of rational singularities (see [12], [13]).

References

[1] E. Artin, "Theory of braids" Ann. of Math. , 48 (1947) pp. 643–649 MR0019087 Zbl 0030.17703
[2] J.S. Birman, "Braids, links and mapping class groups" , Princeton Univ. Press (1974) MR0375281 Zbl 0305.57013
[3] W. Burau, "Ueber Zopfinvarianten" Abh. Math. Sem. Univ. Hamburg , 9 (1932) pp. 117–124 Zbl 0006.03401 Zbl 58.0614.03
[4] A.A. Markov, "Foundations of the algebraic theory of braids" Trudy Mat. Inst. Steklov. , 16 (1945) (In Russian)
[5] B. Gassner, "On braid groups" Abh. Math. Sem. Univ. Hamburg , 25 (1961) pp. 10–22 MR0130309 Zbl 0111.03002
[6] E. Fadell, L. Neuwirth, "Configuration spaces" Math. Scand. , 10 (1962) pp. 111–118 MR0141127 MR0141126 Zbl 0136.44104 Zbl 0122.17803
[7] F.A. Garside, "The braid group and other groups" Quart. J. Math. , 20 : 4 (1969) pp. 235–254 MR0248801 Zbl 0194.03303
[8] D.B. Fuks, "Cohomology of braid groups mod 2" Funktional. Anal. i Prilozhen. , 4 : 2 (1970) pp. 62–73 (In Russian) MR274463
[9] V.I. Arnol'd, "On cohomology classes of algebraic functions that are preserved under Tschirnhausen transformations" Funktional. Anal. i Prilozhen. , 1 (1970) pp. 84–85 (In Russian)
[10] E.A. Gorin, V.Ya. Lin, "Algebraic equations with continuous coefficients and some problems in the theory of braids" Mat. Sb. , 78 (1969) pp. 579–610 (In Russian)
[11] V.I. Arnol'd, "On certain topological invariants of algebraic functions" Trudy Moskov. Mat. Obshch. , 21 (1970) pp. 27–46 (In Russian)
[12] E. Brieskorn, Matematika , 18 : 3 (1974) pp. 46–59 MR0360085
[13] E. Brieskorn, K. Saito, "Artin Gruppen und Coxeter Gruppen" Invent. Math. , 17 (1972) pp. 245–271 MR0323910 Zbl 0243.20037
[14] P. Deligne, "Les immeubles des groupes de tresses généralisés" Invent. Math. , 17 : 4 (1972) pp. 273–302 MR0422673 Zbl 0238.20034
[15] V.Ya. Lin, "Representations of permutation braids" Uspekhi Mat. Nauk. , 29 : 1 (1974) pp. 173–174 (In Russian)
[16] V.I. Arnol'd, "The ring of cohomology groups of crossed braids" Mat. Zametki , 5 : 2 (1969) pp. 227–231 (In Russian)
[17] V.Ya. Lin, "Artin braids and related groups and spaces" Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 17 (1979) pp. 159–227 (In Russian) MR0584570


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Braid theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Braid_theory&oldid=18559
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article