Minimal intersection number
of curves on surfaces.
1
Preliminaries
• Fix an oriented surface S.
• Denote by π the fundamental group of S
• Denote by π* the set of conjugacy classes
of elements of π.
• Denote by V(π*) the module generated by
π* (coefficients in Z or R)
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The Goldman bracket (Goldman, 86)
[ , ]:V(π*) x V(π*) →V(π*)
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The Goldman bracket (Goldman, 86)
[ , ]:V(π*) x V(π*) →V(π*)
[b1 b2 , x]=
- b1 b2 x + b2 b1 x
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The Goldman bracket cont.
[b1 b2 , x]=
b2 b1 x
- b1 b2 x
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Theorem (Goldman, 86)
• The bracket is well defined.
• It is skew-symmetric [a,b]=-[b,a]
• Satisfies the identity,
[[a,b],c] + [[b,c],a] + [[c,a],b] = 0
In other words: V(π*), the vector space generated
by free homot. classes of curves on an orientable
surface has a Lie algebra structure.
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The Lie bracket is a refinement of
the intersection product of curves.
- b1 b 2 x
+ b2 b1 x
Intersection number = 1-1=0
Questions: How good is it this refinement?
Does it really identify all non-removable
intersection points? Or is there
cancellation?
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Is the number of terms of the bracket
counted with multiplicity, equal to the
minimum intersection number of a pair
of conjugacy classes of curves?
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Remark
• (number of terms of the bracket, with mult)
≤
(minimal number of intersection points)
• If there are two representatives of a and b
such that the bracket has no cancellation
(number of terms of the [a,b], with mult)
≥
(minimal number of intersection points)
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The bracket of essentially
intersecting pairs of curves can be
zero
[aab,ab]= - aab ba + baa ab=0
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Theorem (Goldman, 86)
If the bracket of two conjugacy classes of
curves is zero and one of the classes
contains a simple representative, then the
classes have disjoint representatives.
Briefly,
if [a, b]=0 and a simple then a, b disjoint.
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When one of the curves a, b in
[a,b] is simple
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x
The fundamental
group of S is
the amalgamated
free product of the
closure of the
connected
components of S \ x,
(x generates the
amalgamating
group)
The separating case.
x
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A parenthesis of Combinatorial
Group Theory
• Let G and H be two groups.
• Let A be a subgroup of G,
• Let B be a subgroup of H,
• Assume σ : A → B is an isomorphism.
Then the free product of G and H
amalgamating the subgroups A and B by S
is the group
(G*H)/N ,
where N is generated by {a (σ(a))-1, a in A}
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Fix a separating curve x in S
•
•
Denote by S1 and S2 the closure of the
connected components of S\x, and by π1
and π2 their fundamental groups.
π “=“ π1 * π2 /N
S1
x
S2
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•
•
A sequence of elements
w1 ,w2.., wn of a free product
with amalgamation of two
groups π1 and π2 is reduced
if each wi is alternatively in
π1 and π2 and no term
belongs to the amalgamating
group.
A sequence of elements
w1 ,w2.., wn is cyclically
reduced if every cyclic
permutation is reduced . .
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Theorem
Any conjugacy class of π contains elements
that can be written as the product of a
cyclically reduced sequence.
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The bracket of a simple separating
curve x and another curve
Write the “other curve” as the product of a
cyclically reduced sequence,
+[ w1w2w3w4 , x ] =
w2w3w4w1x - w3w4w1w2x + w4w1w2w3x - w1w2w3w4x
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The bracket of a simple separating
curve x and another curve
Since wi+1..wn w1..wi x“=“ w1 w2..wi x wi+1..wn
we have
+[w1w2..wn ,x]=
w1xw2.. wn- w1w2x..wn +...+ (-1)n w1 w2..wnx
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• (number of terms of the bracket, with mult)
≤(minimal number of intersection points)
• If there are two representatives of a and b
such that the bracket has no cancellation
(number of terms of the [a,b], with mult)≥
(minimal number of intersection points)
Therefore (min int)=(number terms in [a,b])
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To show that there is no
cancellation
One needs to prove that if i and j are two
positive integers with different parity,
then two elements of the form
w1w2..wix..wn and w1w2..wjx..wn cannot
be conjugate.
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Note
The sequences
w1, w2, .. , wix, ..,wn and w1, w2 ,.. , wjx, .., wn
are cyclically reduced.
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Theorem
If two cyclically reduced
sequences
w1,w2....wn and v1,v2....vn
are conjugate then there
exists an integer k and a
sequence of elements of
the amalgamating group,
c1,c2....cn such that
1
i  k 1 i  k i  k
wi  c
v c
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Apply the theorem to
w1,w2..,wix,..wn and w1,w2..,wjx,..wn
There exists a sequence of elements of the
group generated by x, namely c1,c2....cn
such that,
1
i  k 1
wi x  c
wi  k ci  k
1
j 1
w j k  c w j xc j
1
h  k 1
wh  c
wh  k ch  k if h  i and h  j  k
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Remark
• We showed that if x is simple and nonseparating, and y is any free homotopy
class then the three following numbers are
equal
– Minimal intersection number of x and y
– Number of terms (with multiplicity) of [x,y]
– Number of term of the reduced sequence of y
with respect to x.
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The non-separating case
can be treated with the same methods using
HNN extensions.
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Theorem (C.)
The Goldman bracket of two curves, one of
them simple, has as many terms as the
intersection number.
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Definition
Changing the orientation of curves
determines an involution,
I : π*
π*
which is can be extended to an linear
automorphism of V(π*)
The set of fixed points of this automorphism
is generated by a+I(a), for a in π* . These
generators are in one-to-one
correspondence with unoriented curves.
Denote this set by F(π*)
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Theorem (Goldman, 86)
F(π*) is a sub-Lie algebra of V(π*)
Proof:
• a, b in π*, then I[a,b]=[I(a),I(b)]
Definition: F(π*) is the Lie algebra of
unoriented curves.
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“90% Theorem” (C.)
• The Lie bracket of unoriented curves has
no cancellation if one of the curves is
simple.
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