Local subhomeotopy groups of bounded surfaces

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Vol. 24, No. 4 (2000) 251–255 S0161171200003379 ©Hindawi Publishing Corp.

LOCAL SUBHOMEOTOPY GROUPS OF BOUNDED SURFACES

DAVID J. SPROWS

(Received 2 February 1999 and in revised form 10 May 1999)

Abstract.LetMndenote the 2-dimensional manifold with boundary obtained by remov-ing the interiors ofndisjoint closed disks from a closed 2-manifoldMand letMn,r de-note the manifold obtained by removingr distinct points from the interior ofMn. The subhomeotopy group ofMn,r, denotedHn(Mn,r), is deﬁned to be the group of all isotopy classes (rel∂Mn,r) of homeomorphisms ofMn,rthat are the identity on the boundary. In this paper, we use the isotopy classes of various homeomorphisms ofS2

n+1,r to investi-gate the subgroup ofHn(Mn,r)consisting of those elements that are presented by local homeomorphisms.

Keywords and phrases. Local subhomeotopy group, isotopy class, twist homeomorphism, spin homeomorphism.

2000 Mathematics Subject Classiﬁcation. Primary 57N37.

1. Introduction. Given a 2-dimensional manifoldY with or without boundary, the homeotopy (or mapping class) group ofY, denotedH(Y ), is defined to be the group of all isotopy classes of homeomorphisms ofY (see [3] or [5] for more details). In the case thatY has a boundary with n boundary components, we define the sub-homeotopy group ofY, denotedHn(Y ), to be the group of all isotopy classes (rel∂Y) of homeomorphisms that are the identity on the boundary ofY. That is, both the homeomorphisms and the isotopies must be fixed on the boundary ofY. Note that the term “local homeomorphism” is used to describe a homeomorphism that is the identity for all values except for those in a “local” region within a given closed diskD. This should not be confused with the more standard usage of this term, i.e., a contin-uous functionf:X→Y such that eachxinXpossesses a neighborhoodU mapped byfhomeomorphically onto a neighborhoodf (U)ofx.

LetMndenote the 2-dimensional manifold with boundary obtained by removing the interiors ofndisjoint closed disks from a closed 2-manifoldMand letMn,r denote the manifold obtained by removingr distinct points from the interior ofMn. If we enclose thendisks andr points to be removed fromM in a diskD, then we say a homeomorphism ofMn,rislocalprovided it is the identity onM₌_M₋_Int_(D)_{. For} ex-ample, iffandgare homeomorphisms ofMn,rthat represent the same isotopy class inH1(M), thenf g−1is a local homeomorphism. It is the subgroup ofHn(Mn,r) con-sisting of elements represented by local homeomorphisms that is the main concern of this paper.

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disk inMthat containsDk,1≤k≤m, in its interior. It will be convenient to consider

Das a sphere with the interior of a disk removed. In particular, we will consider the disksD1,...,Dmas each having its center on the equator and consider the boundary of

Dto be the same as the boundary of a diskDm+1that is also centered on the equator. ThusDS2₋_Int_(Dm

+1). Note that under this interpretation, removal of the points

pk,1≤k≤r, and the interiors of the disksDk, r+1≤k≤r+n, fromMtransforms

Dinto a space homeomorphic toS2

n+1,r, i.e., these sphere withrpoints andn+1 open

disks removed.

In the following we will make use of three types of homeomorphisms of manifolds with boundary.Twisthomeomorphisms are deﬁned in detail in [4, 9]. In essence these are homeomorphisms that are deﬁned on annular neighborhoods of simple closed curves by conjugating the homeomorphism of the annulusAgiven byt(r ,θ)=(r ,θ−

2πr ), whereAis parametrized by (r ,θ) with 1≤r≤2 andθa real number mod 2π. Given two pointspandqcontained in the interior of an embedded annulus the corre-spondingdialhomeomorphism essentially “dials”ptoqin a counterclockwise direc-tion. Details are given in [7], but again the basic idea is to obtain dial homeomorphisms by suitably conjugating the homeomorphism ofAgiven byd(r ,θ)=(r ,θ−2π(r−1))

for 1≤r≤1.5 andd(r ,θ)=(r ,θ−2π(2−r ))for 1.5≤r≤2, where we assumep

corresponds to(3/2,0)andqto(3/2,π)in the annulusA.

Finally, each boundary componentC of a 2-manifold with boundary gives rise to aspinhomeomorphism. We deﬁne the spin homeomorphism scorresponding toC

by lettingebe an embedding of the annulusAonto a collar neighborhood ofCthat sendsr=1→Cand settings=ete−1_{, where}_t_{is the twist homeomorphism of}_A_.

Note that all three types of homeomorphisms deﬁned above(t=twist,d=dial,s=

spin)are the identity on the boundary of the annulus and so can be extended by the identity from any embedded annulus to the entire manifold containing this embed-ded annulus.

3. Local homeomorphisms. LetM andDS2₋_Int_(Dm

+1)be as described in the previous section. The ﬁrst local homeomorphisms we will consider are given by twist homeomorphisms. These twist homeomorphisms will correspond to simple closed curves inD, but initially it will be more convenient to describe these curves inS2_{, i.e.,} the space obtained fromDby ﬁlling in the diskDm+1.

For eachiand j with 1≤i < j≤m+1, let αij be a simple closed curve in S2 that is disjoint from the union of the closed disks D1,...,Dm+1 and encloses the disksDi andDj,...,Dm+1. That is, one of the disks bounded byαij inS2_contains

Di and Dj,...,Dm+1 and the other disk bounded by αij contains D1,...,Di−1 and

Di+1,...,Dj−1. In addition, assumeαijis below the disksDi+1,...,Dj−1, i.e., assume the “neck” ofαij is in the southern hemisphere ofS2_{. Since}_αij _{is disjoint from the disk}

Dm+1, we can considerαij as a curve inD⊂M. Letaijbe the twist homeomorphism ofMcorresponding toαij.

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when dealing with such homeomorphisms we will use the same letter denote the given homeomorphism ofMand its restriction toMn,r.

Note that eachaij represents a local homeomorphism ofMn,r. LetH'

n(Mn,r)denote

the group of all isotopy classes inHn(Mn,r)that are represented by local homeomor-phisms, i.e., homeomorphisms ofMn,r that are ﬁxed onM−D. These local homeo-morphisms must be the identity on the boundary component of each of the disks

Dr+1,...,Dm+1, but do not necessarily induce the identity map onF if we ﬁll in each of the pointsp1,...,pr.

In order to deal with local homeomorphisms that may permute the points ofF, we introduce the following class of dial homeomorphisms. As was the case with the home-omorphismsaij, it will be convenient to ﬁrst deﬁne these dial homeomorphisms onS2_. For each pair of adjacent pointspi andpi+1, 1≤i < r, embed an annulus about a simple closed curve passing throughpi andpi+1. Since each of the disksDj, 1≤

j≤m+1, is on the equator ofS2_{, we can assume that this annulus is disjoint from} all of disksDkwith k≠i, i+1 and that the embedding takes the segmentsθ=0 (respectively, θ =π) to a section of the equator containing pi+1 (respectively, pi). Deﬁnedi to be the dial homeomorphism corresponding topi andpi+1. As was the case with theaij, eachdican be considered a homeomorphism ofMn,r.

Next letCidenote the boundary component of the diskDiforr+1≤r≤m+1 and letsibe the corresponding spin homeomorphism.

Given a local homeomorphismhofMn,r we use ¯hto denote the equivalence class ofhinHn(Mn,r). Note that each local homeomorphism is the identity on eachCifor

r+1≤r ≤m+1, but the isotopies needed to determine the equivalence classes in

Hn(Mn,r)are required to be the identity only onCiforr+1≤i≤m. The following theorem shows that every element inH'

n(Mn,r)can be expressed in

terms of equivalence classes of twists, dials and spins. Theorem3.1. H_n(Mn,r' ₎_{is generated by}_T_∪_L_∪_S_{, where}

T= ∪m+1

k=2

¯

aik: 1≤i < k, L=di¯ : 1≤i < r, S=¯si:r+1≤i≤m+1.

(3.1) Moreover, the “commutator” relations among these generators are given as follows:

1. (1) Ifp < qandi > t, then ¯

aipatq¯ a¯−1

ip =

¯

aiq¯batq¯ aiq¯ b¯−1, where¯b=apq¯ ···aq¯−1,q.

(2) Ifp < qandi < t < p, then ¯

aipatq¯ a¯−1

ip = _¯

baiq¯ atq¯ b¯aiq¯ −1.

(3) ¯aipatq¯ a¯−1

ip =atq¯ in all remaining cases.

2. (1) ¯didk¯d¯−1

i =d¯−1k di¯dk,¯ if|i−k| =1.

(2) ¯didk¯d¯−1

i =dk,¯ if|i−k| ≥2.

3. (1) ¯dpajk¯ d¯−1

p =aj¯+1,k,ifj < k−1andp=j. (2) ¯dpajk¯ d¯−1

p =a¯−1jkaj¯−1,ka¯−1jk,ifp=j−1.

(3) ¯dpajk¯ d¯−1

p =aj,k¯ −1a¯−1jkak¯ −1,k,ifk=r andj < k−1andp=r−1. (4) ¯dpajk¯ d¯−1

p = aj,k¯ −1aj,k¯ +1a¯−1jk

¯

ak−1,k+1ak,k¯ +1a¯k−1+1,k+2a¯−1k−1,k

,ifp = k−1and

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(5) ¯dpajk¯ d¯−1

p =ajk¯ for all remaining choices ofj,k,p.

4. (1) ¯sic¯¯s−1

i =c¯any generatorc¯.

The relations given above are expressed in the form that naturally occurs when deriving these relations inH'

n(Mn,r), but note that each of these relations can be

written with the commutator of the given generators on the left by multiplying through by the inverse of the appropriate generator.

Proof of Theorem3.1. LetGbe the subgroup ofH'

n(Mn,r)consisting of all those

elements that are represented by local homeomorphisms ofMn,rthat induce the iden-tity on the setF when we ﬁll in the pointsp1,...,pr. The inclusion map gives rise to the short exact sequence (1) 1→G→H'

n(Mn,r)→B→1, whereBis isomorphic to the

symmetric groupSp. SinceSpis generated by the set of all transpositions of the form

(i,i+1), it follows thatLis a generating set forB.

To ﬁnd a set of generators forGwe ﬁrst consider the correspondence between local homeomorphisms ofMn,rand homeomorphisms ofS2

n+1,rthat is obtained by viewing DasS2₋_Int_(Dm

+1). In particular, if we letGdenote the subgroup ofHn+1(Sn2+1,r)

consisting of all elements that are represented by homeomorphisms that induce the identity onF, then there is a natural homeomorphism fromG_onto_G_{that sends the} isotopy class (rel∂S2

n+1,r) of a homeomorphismhinG, denotedh, to the isotopy

class (rel∂Mn,r) ofhinG, denoted ¯h. Note that since this homeomorphism is onto, any generating set forG_{gives rise to a generating set (with possible redundancies)} forG.

LetI(respectively,I_{) denote the subgroup of}_G_{(respectively,}_G_{) consisting of those} elements that are represented by homeomorphism that are isotopic to the identity. That is, a non-trivial element ofI(respectively,I₎_{is isotopic to the identity, but not} by an isotopy that is ﬁxed on∂Mn,r (respectively,∂S2

n+1,r).

The inclusion map induces a short exact sequence (2) 1→I_→_G σ_→_H_→_{1, where}

H_{is obtained by identifying two elements of}_G_{provided they are isotopic. That is,}

σ (g₎₌_{σ (h}₎_provided_g _{is isotopic to}_h_{, where the isotopy is not required to be} ﬁxed on the boundary.

Since rotations are essentially the only isotopies of the circle that start and end with the identity (see [8]), a generating set forI_{is given by any set of homeomorphisms} ofS2

n+1,r that give all possible rotations of eachci, r+1≤r≤m+1. In particular,I

is generated by the setS_{= {}_s

i:r+1≤i≤m+1}.

A generating set forH_{can be obtained by making use of [6, Theorem 4.1] and the} fact that all the homeomorphisms and isotopies given in this theorem can be taken to be fixed on closed disks that contain each of the points involved in the fixed point set. In particular, if we take{p1,...,pm+1}as the fixed point set and useDr+1,...,Dm+1 as the corresponding disks about the pointspr+1,...,pm+1, then it follows thatHis generated by the setT_{consisting of those elements that are represented by the twist} homeomorphismsaik,1≤i < k, k=2,...,m+1.

Using the short exact sequence (2), we have thatG_{is generated by}_T_∪_S_{. Using} the short exact sequence (1), the relationship betweenG_and_G_{, and the fact that}_B_is generated byL, we have thatGis generated byT∪L∪S.

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relations that hold for the transpositions (i,i+1) is the symmetric groupSr (see [1]). The relations given in 4 follow from the fact that the support of eachsican be taken to be disjoint from the supports of the dial and twist homeomorphisms.

The relations in 1 and 3 are determined with the help of two properties of twist homeomorphisms. One useful property is that ifαandβare simple closed curves in the interior ofMn,randαis isotopic toβ(rel∂Mn,r), then the twist homeomorphisms corresponding to these curves are isotopic(rel∂Mn,r). Another useful property deals with the twist homeomorphism corresponding toh(α), wherehis a homeomorphism ofMn,r andαis a simple closed curve in the interior ofMn,r. If we letadenote the twist homeomorphism corresponding toα, then the twist homeomorphism given by

h(α)is isotopichah−1_{. Moreover, if}_h _{is ﬁxed on a boundary component}_Ci_{, then} we can assume this isotopy also leaves the boundary component ofCi ﬁxed. Using these two properties of twist homeomorphisms, together with the techniques de-scribed in [6] establishes the relations in 1 and 3 and hence completes the proof of the theorem.

Usually the relations in the theorem do not yield a recognizable group. A description of the group obtained for various values ofmandnin the case whenM=S2_{can be} found in [6, 7]. Note that, depending on the choice of manifoldM and the value of

n, it may be possible to reduce the size of the generating set given in the theorem. For example, ifM=S2_{, then the homeomorphism}_sm

+1is isotopic (rel∂Mn,r) to the identity, so that the generator ¯sm+1is redundant in this case. Similarly, the value of

nhas an impact on the relations that hold among these generators. In particular, if

n≥3, then relations such as those discussed in [2, 8] will hold inH'_(Mn,r)_.

References

[1] H. S. M. Coxeter and W. O. J. Moser,Generators and Relations for Discrete Groups, Springer-Verlag, Berlin, 1965. MR 30#4818. Zbl 133.28002.

[2] D. L. Johnson,Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc.75(1979), no. 1, 119–125. MR 80h:57008. Zbl 407.57003.

[3] J. P. Lee,Homeotopy groups of orientable2-manifolds, Fund. Math.77(1972), no. 2, 115– 124. MR 47#9625. Zbl 241.57020.

[4] W. B. R. Lickorish,A ﬁnite set of generators for the homeotopy group of a2-manifold, Proc. Cambridge Philos. Soc.60(1964), 769–778. MR 30#1500. Zbl 131.20801.

[5] G. S. McCarty, Jr., Homeotopy groups, Trans. Amer. Math. Soc. 106(1963), 293–304. MR 26#3062. Zbl 113.17201.

[6] D. J. Sprows,Subhomeotopy groups of the2-sphere withnholes, Fund. Math.108(1980), no. 1, 1–5. MR 81m:57026. Zbl 432.57002.

[7] ,Homeotopy groups of punctured spheres with holes, Fund. Math.115(1983), no. 3, 207–212. MR 85c:57037. Zbl 521.57007.

[8] ,A note on the presentations of isotopy groups of bounded2-manifolds, Kumamoto J. Math.2(1989), 28–31. MR 90j:57001. Zbl 684.57007.

[9] H. Zieschang, E. Vogt, and H. Coldewey, Surfaces and Planar Discontinuous Groups, Lecture Notes in Mathematics, vol. 835, Springer, Berlin, 1980. MR 82h:57002. Zbl 438.57001.

David J. Sprows: Department of Mathematical Sciences, Villanova University, Vil-lanova, PA19085, USA