A SURGICAL PROOF OF THE CONTRACTIBILITY OF THE
COMPLEX OF MINIMIZING CYCLES
We give a short proof of the ontratibility of the omplex of minimizing yles.
Our proof is inspired by Hather's proof of the ontratibility of the ar omplex for
a surfae [2℄. This theorem was used by the authors to ompute the ohomologial
dimension of the Torelli group for a surfae [1℄. We start by briey realling the
denition of the omplex of minimizing yles.
Curves and yles. Let S = Sg be a losed onneted orientable surfae of genus
g. A 1-yle in S is a nite formal sum
X
ki i
where ki 2 R, and eah i is an oriented simple losed urve in S . The set fi :
ki 6= 0g is alled the support of the 1-yle. We say that the 1-yle is simple if the
urves of the support are pairwise disjoint, and we say that it is positive if eah ki
is positive.
P
A basi yle is a simple positive 1-yle ki i with the property that its support
forms a linearly independent subset of H1 (S; R).
Let S denote the set of isotopy lasses of oriented simple losed urves in S . There
is a natural set map from the set of 1-yles in S to RS .
A multiurve in the surfae S is a nonempty olletion of disjoint simple losed
urves in S that are isotopially nontrivial and isotopially distint.
The rst step is to x an arbitrary nontrivial
element x of H1 (S; Z). Next, let M be the set of isotopy lasses of oriented multiurves M in S with the property that eah oriented urve of M appears in the
support of some basi yle for x supported in M . The set M has a natural partial
ordering under inlusion (the orientation is important here). As suh, we an think
of M as a ategory where the morphisms are inlusions.
Constrution of the omplex.
Given M 2 M, let PM be the polytope in RS given by the onvex hull of the images
of the basi yles for x supported (with orientation) in M . The olletion fPM g
is a ategory under inlusion, whih we denote F . The funtor given by
M
7! PM
is an isomorphism from M to F . We dene the omplex of minimizing yles B (S )
as the geometri realization of the olimit of this funtor:
B(S ) =
M 2M
olim fPM g :
1
2A SURGICAL PROOF OF THE CONTRACTIBILITY OF THE COMPLEX OF MINIMIZING CYCLES
~ be the set of isotopy lasses of oriented multiurves
Let M
M in S with the property that eah oriented urve of M lies in the support of some
1-yle that represents x and is supported in M . If in the denition of B (S ) we
~ , we get a spae, whih we denote B~(S ), that ontains B (S ).
replae M with M
The spae B~(S ) has the struture of a ell omplex, exept that some of the \ells"
are not ompat.
An auxiliary omplex.
A haraterization of B(S ). Below, we give two alternate haraterizations of
P
points of B (S ). For a positive 1-yle = ki i , an equation of the form [i1 ℄ +
+ [in ℄ = 0 (n > 0) is alled a one-sided relation, and we may say that satises
a one-sided relation. Also, we say that is a minimizing yle for x if there is a
hyperboli metri on S for whih realizes the
minimum length over all 1-yles in
P
S representing x (the length of is the sum ki `(i ), where `(i ) is the length of
the unique geodesi homotopi to i ).
Lemma 1.
Let be a 1-yle in S . The following are equivalent.
(1) represents a point of B (S ).
(2) is a minimizing yle for x.
(3) satises no one-sided relations.
Proof. The equivalene of (1) and (2) is the ontent of Lemma 3.12 in our original
paper [1℄.
That (2) implies (3) is straightforward: if satises a one-sided relation, say,
[1 ℄ + [n ℄ = 0, then we may shorten by subtrating from the oeÆients of
1 ; : : : ; n , where is any positive number smaller than the minimum of fk1 ; : : : ; kn g.
To show that (3) implies (1), we suppose that satises no one-sided relations. The
goal is to show that there is a basi yle for x supported in fi g with some xed
i0 in the support. If is not already basi, that means S [i is disonneted. If
S 0 is one omponent, then the i that form the boundary of S 0 satisfy a relation of
the form
[1 ℄ + + [p ℄ = [p+1 ℄ + + [p+q ℄
with p; q > 0. Suppose without loss of generality that i0 > p and that k1 is
the minimum of fk1 ; : : : ; kp g. The following is a new simple positive 1-yle that
represents x:
= ((k2 k1 )2 + (kp k1 )p )+((kp+1 +k1 )p+1 + (kp+q +k1 )p+q )+
0
X
i>p+q
ki i :
The 1-yle 0 is a simple positive 1-yle that ontains i0 in its support. What is
more, S 0 has one fewer onneted omponent than S . By indution, we an
nd a 1-yle representing x that is supported in fi g and that is nonseparating in
S . Suh a 1-yle is basi.
A SURGICAL PROOF OF THE CONTRACTIBILITY OF THE COMPLEX OF MINIMIZING CYCLES3
Suppose that d and d0 are two 1-yles in S representing
points of B (S ), assume that d and d0 have the minimal number of intersetions in
their isotopy lasses, and let be the 1-yle td + (1 t)d0 , where t 2 [0; 1℄; note
that represents the lass x. In general, is not simple. We now explain how to
do surgery to onvert into a simple 1-yle Surger(), whih is anonial up to
isotopy.
Surgery on 1-yles.
By Poinare Duality, the homology lass x orresponds to a ohomology lass, and
hene to a homotopy lass of based maps from S to the irle S 1 . We will use the
yle to onstrut an expliit representative ' of this homotopy lass.
P
As above, say that =
ki i . We thiken eah i to a band (annulus) Ai =
S 1 [0; ki ℄. We hoose the Ai and oordinates (i ; ti ) so that the following onditions
hold (f. the left hand side of Figure 1).
(1)
(2)
(3)
(4)
(5)
i is oriented in the positive diretion.
The intersetion of j with Ai (i 6= j ) is a oordinate ar.
The intersetion of two annuli is a oordinate retangle in both annuli.
The retangles of intersetion are disjoint.
In eah retangle of intersetion, an ar is a oordinate ar in one annulus
if and only if it is a oordinate ar in the other.
We an now deneRa measure i on the set of ars in Ai . For an ar : [0; 1℄ ! Ai ,
we dene i () = dti . We get a measure i on ars in S by adding the measures
of the omponents ontained in Ai . Finally, we obtain a measure on ars in S
given by
X
() = i ():
For a losed loop , we see that () is the same as the algebrai intersetion of with .
We are now ready to dene the map ' : S ! S 1 . Let p be an arbitrarily hosen
basepoint of S , let q be an arbitrary point of S , and let q : [0; 1℄ ! S be an
arbitrary path from p to q . Thinking of S 1 as R=Z we dene '(q ) as the frational
part of (q ). Sine x is an integral homology lass, this map is well-dened.
We now use the map ' : S ! S 1 to dene the simple 1-yle Surger(). The
preimage of eah regular point of ' in S 1 is a 1-manifold, that is, a olletion of
pairwise disjoint urves in S . If we onsider the union of the preimages of regular
points in S 1 all at one, we see some number of parallel families of urves. Eah
family has a well-dened thikness, oming from the measure on the irle (we use
the measure oming from R). As suh, we replae eah family of urves with a single
weighted urve. Also, we disard any urves that are homotopi to a point. The
resulting 1-hain is the desired simple 1-hain Surger(). From the onstrution,
we see that Surger() orresponds to the same ohomology lass as (integrating
against Surger() gives a map homotopi to '), and so Surger() represents the
homology lass x.
All of the hoies made were unique up to isotopy, and so the 1-yle Surger() is
a well-dened point of B~(S ).
4A SURGICAL PROOF OF THE CONTRACTIBILITY OF THE COMPLEX OF MINIMIZING CYCLES
In what follows, we will use the fat that, for any xed 1-yle oming from a
point of B (S ), the funtion
d 7! Surger(t + (1 t)d)
is a ontinuous map from B (S ) [0; 1℄ to B~(S ). For ontinuity, the key point is
that the ondition that some isotopy lass is in the support of Surger(t + (1 t)d)
is an open ondition.
kj
kj
ki
ki
kj
PSfrag replaements
kj
Figure 1.
Surgery on 1-yles
The onstrution of Surger() is more easily explained by a single piture; see
Figure 1. The well-denedness is easy to see in this gure. What is not easy to see
in the gure is that Surger() is a 1-yle. For instane, onsider on the torus two
urves and that interset one, and let d be the 1-yle + (1 ) where
is irrational. If we perform the surgery indiated in Figure 1 at the intersetion,
the result is not a well-dened 1-yle. The point is that d does not represent a
rational homology lass. In our onstrution of Surger() it was essential that x
was an integral lass (a slight modiation is needed in the rational ase).
The result of the surgery dened above is always a simple
positive 1-yle representing x. However, it is not neessarily true that this 1-yle
represents a point of B (S ); rather, this 1-yle is in general a point in B~(S ). To
address this issue, we now dene a strong deformation retration Drain : B~(S )
! B(S ).
Draining 1-yles.
As a basi example, suppose that is the formal sum of a 1-yle representing a
point of B (S ) with a weighted oriented separating urve d. The yle Drain(),
in this ase, is obtained by \draining" the oeÆient of d to 0 (we think of the
subsurfae bounded by d as the drain).
We now return to the general ase; let be a 1-yle representing a point of B~(S ).
Let fSi g be the set of embedded subsurfaes of S that have boundary in . Say
that fS1 ; : : : ; Sk g is the subset of fSi g onsisting of subsurfaes that give rise to
1-sided relations. Suppose that the 1-sided relation orresponding to S1 is
[1 ℄ + + [p ℄ = 0:
For small enough , we obtain a new 1-yle representing x by starting with and
dereasing eah of the oeÆients of 1 ; : : : ; p by .
A SURGICAL PROOF OF THE CONTRACTIBILITY OF THE COMPLEX OF MINIMIZING CYCLES5
We may simultaneously onsider all of the k one-sided relations at one: again, we
start with , and we derease the oeÆient of i by n, where 0 n k is the
number of one-sided relations in whih i appears.
We adjust weights in this way, until some oeÆient beomes 0. At this point, we
repeat the proess, starting with the new 1-yle. When there are no more 1-sided
relations, we stop. The resulting 1-yle is alled Drain(). By Lemma 1, Drain()
represents a point of B (S ). Note in partiular, that, sine no arbitrary hoies were
made, the 1-yle Drain() is well-dened.
The way we dened the Drain map, it is a strong deformation retration of B~(S )
onto B (S ). Below, it will suÆe to think of Drain as a ontinuous map B~(S )
! B(S ).
The proof of ontratibility. We hoose an arbitrary basepoint v of B (S ). We
will now dene a deformation retration of B (S ) to the point v .
Denote by a 1-yle representing v . The retration H : B (S ) [0; 1℄ ! B (S ) is
given by
H (w; t) = Drain(Surger(t + (1 t)d))
where w is an arbitrary point of B (S ), and d is a representative 1-yle.
The map H is a strong deformation retration of
B(S ) is ontratible.
B(S ) onto the point v, and so
Aknowledgements. This work was in part motivated by disussions with Tara
Brendle and Karen Vogtmann. We would like to thank Chris Leininger for suggesting a key improvement, and Allen Hather for pointing out a gap in an earlier
draft.
Referenes
[1℄ Mladen Bestvina, Kai-Uwe Bux, and Dan Margalit. The dimension of the Torelli group.
arXiv:0709.0287.
[2℄ Allen Hather. On triangulations of surfaes. Topology Appl., 40(2):189{194, 1991.