First cohomology of pure mapping class groups of big genus one and zero surfaces
FFirst cohomology of pure mapping classgroups of big genus one and zero surfaces
George Domat and Paul Plummer
Abstract.
We prove that the first integral cohomology of pure map-ping class groups of infinite type genus one surfaces is trivial. Forgenus zero surfaces we prove that not every homomorphism to Z fac-tors through a sphere with finitely many punctures. In fact, we get anuncountable family of such maps. Contents
1. Introduction 12. Background 32.1. Big pure mapping class group 32.2. Polish groups and automatic continuity 43. Genus one 53.1. Pure mapping class group of finite type genus one surfaces 53.2. Homology of exhaustions 63.3. Proof of theorem 1.1 74. Genus zero 9References 12
1. Introduction
Previous work in [APV17] describes the first integral cohomology of puremapping class groups of infinite type surfaces of genus at least 2 in termsof ends accumulated by genus. They prove that the cohomology of thecompactly supported pure mapping class group is trivial and that the onlyother homomorphisms come from handleshifts. In the finite type setting forsurfaces without boundary it is known that once the genus is at least onethe integral cohomology of the pure mapping class group is trivial. For a
Mathematics Subject Classification.
Key words and phrases. infinite-type surface; big mapping class groups; group coho-mology; polish groups; automatic continuity.Domat was partially supported by NSF DMS-1607236 amd NSF DMS-1840190.Plummer was partially supported by NSF DMS-1651963 and NSF DMS-1611758. a r X i v : . [ m a t h . G T ] F e b GEORGE DOMAT AND PAUL PLUMMER sphere with finitely many punctures the rank of the first cohomology is afunction of the number of punctures.We investigate the integral cohomology in the infinite type genus one andzero cases. In the genus one case we prove the following.
Theorem 1.1.
Let S be a genus one surface without boundary and with anonempty closed set of marked points, representing punctures. Let PMCG( S ) be the pure mapping class group of S . Then H (PMCG( S ); Z ) = 0 . When S is of finite type this is well known [FM11], so we will focus onthe case when S is not of finite type.The idea of the proof is that if we have a homomorphism to Z which is notzero, then it is non-zero on some Dehn twist about a separating curve. Inturn this will imply it is non-zero on a sequence of Dehn twists whose productconverges in the group but whose image in Z will not converge. Work of[APV17] and [D61] shows that any homomorphism to Z has to be continuous.We will use this sequence of Dehn twists to contradict the continuity of thehomomorphism. We make use of the Gervais star presentation [G01] to seehow the homology groups of a finite type exhaustion fit together.For surfaces of genus zero we have forgetful maps to finite type puncturedspheres. The first integral cohomology of pure mapping class groups ofspheres with at least four punctures is nontrivial so we cannot hope for asimilar result as in Theorem 1.1 for the genus zero case. However, we canask: Do all homomorphisms from the pure mapping class group of an infinitetype genus zero surface to Z factor through a forgetful map to a sphere withfinitely many punctures? We answer the question negatively by constructinga specific homomorphism for the flute surface which does not factor as such.When S is of infinite type and genus 0 there is always a forgetful map tothe flute surface. The specifics of the construction then lead to the followingtheorem: Theorem 1.2.
Let S be a genus zero infinite type surface. Then the integralcohomology group H (PMCG( S ); Z ) contains cohomology classes which donot come from forgetful maps to finite type genus zero surfaces. Furthermore,there is an uncountable family of such classes. The uncountability comes from being able to vary the construction toencode a Cantor set of cohomology classes.This section of the paper will rely on the fact that any finite type subsur-face of a genus zero surface has mapping class group isomorphic to a braidgroup. [F06] contains some results and discussions on infinite stranded braidgroups and its connections to the mapping class group of a disk with infin-itely many punctures.Our results together with those from [APV17] give an almost completepicture of the first integral cohomology of these big pure mapping class
OHOMOLOGY OF MCG OF GENUS ONE AND ZERO SURFACES 3 groups. The only piece still missing is an explicit description of the coho-mology in the genus zero case. The cohomology groups break down into thefollowing categories: • S has more than one end accumulated by genus ([APV17]): Therank of H (PMCG( S ); Z ) is one less than the number of ends accu-mulated by genus if there are finitely many such ends and infiniteif there are infinitely many. Furthermore, all non-trivial classes arenot supported on finite type subsurfaces. • S has at most one end accumulated by genus with g > H (PMCG( S ); Z ) is trivial. • S is genus zero (Theorem 1.2): H (PMCG( S ); Z ) has infinite rankand contains classes both supported on finite type subsurfaces andones that are not. Question.
What is an explicit description of H (PMCG( S ); Z ) when S isgenus zero?Throughout the paper we will identify H ( G ; Z ) with the set of homo-morphisms G → Z . We will also usually be considering punctures as markedpoints and conflating compact subsurfaces with finite type subsurfaces forthe sake of convenience.
Acknowledgments:
We thank Mladen Bestvina and Jing Tao for nu-merous helpful conversations throughout this project. We also thank JaviarAramayona, Priyam Patel, and Nick Vlamis for their interest and commentson an earlier draft. We thank the referree for numerous helpful comments.This work was started at the Fields Institute’s Thematic Program on Te-ichm¨uller Theory and its Connections to Geometry, Topology and Dynamicsand as such we thank the institute and the organizers for their support.
2. Background
Let S be a connected, orientable,second-countable surface, possibly with boundary. Let Homeo + ∂ ( S ) be thegroup of orientation preserving homeomorphisms of S which fix the bound-ary pointwise. The mapping class group , MCG( S ), is defined to beMCG( S ) = Homeo + ∂ ( S ) / ∼ where two homeomorphisms are equivalent if they are isotopic relative to theboundary of S . Homeo + ∂ ( S ) is equipped with the compact-open topology,which induces the quotient topology on MCG( S ). Subgroups of MCG( S )come equipped with the subspace topology.The pure mapping class group , PMCG( S ), is defined to be the kernelof the action of MCG( S ) on the space of ends of S .We say f ∈ MCG( S ) is compactly supported if f has a representativethat is the identity outside of a compact set in S . We denote the subgroupof MCG( S ) of compactly supported mapping classes as PMCG c ( S ). Note GEORGE DOMAT AND PAUL PLUMMER that any compactly supported mapping class is automatically in PMCG( S ).Patel and Vlamis proved Theorem 2.1 ([PV17]) . PMCG c ( S ) = PMCG( S ) if and only if S has atmost one end accumulated by genus. In particular, PMCG c ( S ) = PMCG( S ) if S is genus one or zero. We say that a sequence of curves { α i } leaves every compact set if forevery compact subset K of S there exists an N > α n ∩ K = ∅ for all n > N . A Polish group is a topological group that is separableand completely metrizable as a topological space.
Lemma 2.3 ([APV17], Corollary 2.5) . The group
MCG ± ( S ) is a Polishgroup with the compact-open topology. Hence PMCG( S ) is a Polish groupas it is a closed subgroup. Remark 2.4.
The above is using work in [HMV17] and [BDR17] whichshows that the automorphism group of the curve graph of S is isomorphicto MCG ± ( S ).This is important for our applications because of the following theorem ofDudley. Dudley’s theorem is more general but we state the relevant versionof the theorem to our work. Theorem 2.5 ([D61]) . Every homomorphism from a Polish group to Z iscontinuous. We can use these results to obtain the following necessary condition for amap from PMCG( S ) to Z to be a homomorphism. Lemma 2.6.
For any surface S and homomorphism f : PMCG( S ) → Z , f cannot be non-zero on a sequence of Dehn twists about curves that leaveevery compact (or finite type) subsurface of S . In other words, there is acompact (or finite type) subsurface K ⊂ S such that if f ( T α ) (cid:54) = 0 then α ∩ K (cid:54) = ∅ where T α is the Dehn twist about the curve α . Proof.
The statement is trivial for finite type surfaces. By Lemma 2.3 andTheorem 2.5 f is continuous. Suppose f were nonzero on such a sequence ofDehn twists. Then we could find a sequence of Dehn twists { T α i } such that f ( T α i ) > i (possibly taking inverses if all twists are negative) andthe α i leave every compact set. Then (cid:81) ni =1 T α i converges in PMCG( S ) as n goes to infinity. However, f ( (cid:81) ni =1 T α i ) = (cid:80) ni =1 f ( T α i ) does not converge in Z , contradicting the continuity of f . (cid:3) We also get the following immediate consequence.
OHOMOLOGY OF MCG OF GENUS ONE AND ZERO SURFACES 5 c c c d d d b Figure 1.
Curves in a Star Relation.
Proposition 2.7.
Let S be an infinite type surface with at most one endaccumulated by genus. Let K ⊂ K ⊂ · · · be an exhaustion of S by finitetype surfaces. These induce inclusions PMCG( K n ) (cid:44) → PMCG( K n +1 ) forall n . Then we have that H (PMCG( S ); Z ) injects into the inverse limit lim ←− H (PMCG( K n ); Z ) . Proof.
We get a map H (PMCG( S ); Z ) → lim ←− H (PMCG( K n ); Z ) by re-striction of a cohomology class to the subsurfaces K n . Now suppose that φ ∈ H (PMCG( S ); Z ) restricts to the zero map on each PMCG( K n ). Givenany f ∈ PMCG( S ) we can approximate f by a sequence of compactly sup-ported mapping classes by Theorem 2.1. Thus by continuity of φ and The-orems 2.3 and 2.5, we see that φ is the zero map on all of PMCG( S ). (cid:3)
3. Genus one
Let S b ,n be a surface of genus 1 with n punctures and b boundary compo-nents. One of the main tools we use is a description of PMCG( S b ,n ) in termsof Gervais star relations.For any subsurface of S b ,n homeomorphic to S and curves as in Figure1 we have the relation ( T c T c T c T b ) = T d T d T d , where T a denotes theDehn twist about the curve a . This relation is called a star relation . Wehave degenerate star relations when one or more of the boundary curves, d i , is null-homotopic in S b ,n , e.g., if d is null-homotopic then the relationbecomes ( T c T c T c T b ) = T d T d . In fact, a presentation of the mappingclass group can be defined using these star relations and braid relations.Such a presentation can be found in [G01].We will also be using the first homology and cohomology of PMCG( S b ,n ). GEORGE DOMAT AND PAUL PLUMMER
Theorem 3.1 ([K02]) . H (PMCG( S b ,n ); Z ) ∼ = Z b when b > . Furthermore,we can choose a basis for the homology corresponding to b − Dehn twistsabout b − boundary components and a Dehn twist about a nonseparatingcurve. When b = 0 we have H (PMCG( S ,n ); Z ) = Z / Z . Note that by an application of the change of coordinates principle allDehn twists about nonseparating curves are identified in homology. Whenthe surface does not have boundary the homology can be derived directlyfrom a degenerate star relation.
Let S be any orientable infinite typegenus 1 surface and let F ⊂ F ⊂ F ⊂ · · · be an exhaustion of S by finitetype subsurfaces with F having genus one. Without loss of generality wemay assume that F n +1 is obtained from F n by gluing a disk with a puncture,an annulus with a puncture, or a pair of pants to some boundary curve of F n . Let P n = PMCG( F n ).Note that PMCG c ( S ) = lim −→ P n , so any homomorphism f : PMCG c ( S ) → Z restricts to a map f n : P n → Z for all n . Conversely, we say that a sequenceof maps { f n : P n → Z } is consistent if f n +1 | P n = f n for all n . If we defineconsistent maps f n for each n , these determine a map f in the limit.A basis for H ( P n ; Z ) is given by (cid:104) τ, ∂ , . . . , ∂ m − (cid:105) where τ represents aDehn twist about a non-separating curve and ∂ i represents a Dehn twistabout a boundary curve b i , where all the boundary curves are b , . . . , b m .We now describe how the exhaustion is realized in homology. Lemma 3.2.
The map H ( P n ; Z ) → H ( P n +1 ; Z ) takes one of the followingthree forms depending on how F n +1 is obtained from F n :(1) (Disk with a Puncture); If we cap a boundary curve b i with a punc-tured disk then ∂ i is killed and our new basis for H ( P n +1 ; Z ) is givenby (cid:104) τ, ∂ , . . . , ∂ i − , ∂ i +1 . . . , ∂ m − (cid:105) .(2) (Annulus with a Puncture): If we glue an annulus with a punctureto b i then we get a new boundary component b (cid:48) i and ∂ i = ∂ (cid:48) i ; i.e. thetwo Dehn twists will be homologous. A new basis for H ( P n +1 ; Z ) is (cid:104) τ, ∂ , . . . , ∂ i − , ∂ (cid:48) i , ∂ i +1 , . . . , ∂ m − (cid:105) .(3) (Pair of Pants): If we glue a pair of pants to b i then we get two newboundary components, say b i and b i , and ∂ i = ∂ i + ∂ i . The newbasis for H ( P n +1 ; Z ) is (cid:104) τ, ∂ , . . . ∂ i − , ∂ i , ∂ i , ∂ i +1 , . . . , ∂ m − (cid:105) Proof.
See Figure 2 for examples of the annulus with a puncture and pairof pants cases.(1) Suppose F n +1 is obtained from F n by capping the boundary curve b i with a punctured disk. Then b i becomes null homotopic in F n +1 so ∂ i is now trivial. Furthermore, the rank of H ( P n +1 ; Z ) is decreasedby one and (cid:104) τ, ∂ , . . . , ∂ i − , ∂ i +1 . . . ∂ m − (cid:105) is a basis. OHOMOLOGY OF MCG OF GENUS ONE AND ZERO SURFACES 7 (2) Suppose F n +1 is obtained from F n by gluing an annulus with a punc-ture to b i . Let b (cid:48) i be the new boundary curve and ∂ (cid:48) i the Dehn twistabout it. Now we will apply a star relation to see that ∂ i = ∂ (cid:48) i . Let a be the curve on F n +1 that bounds all boundary components andpunctures other than b (cid:48) i and the puncture we glued on. Then wehave two degenerate star relations 12 τ = α + ∂ (cid:48) i and 12 τ = α + ∂ i where α is the Dehn twist about a . Thus we see that ∂ (cid:48) i = ∂ i asdesired.(3) Suppose F n +1 is obtained from F n by gluing a pair of pants to b i . Let b i and b i be the two new boundary curves and ∂ i and ∂ i the Dehntwists about them respectively. We proceed as in the previous case.Let a be the curve on F n +1 that bounds all punctures and boundarycurves other than b i and b i . Then we have one degenerate starrelation 12 τ = α + ∂ i and a star relation 12 τ = α + ∂ i + ∂ i . Thuswe see that ∂ i = ∂ i + ∂ i . (cid:3) We also need the following relation in the homology of these finite typesurfaces.
Lemma 3.3. If F is a genus one surface of finite type with boundary curves b , . . . , b m then in H (PMCG( F ); Z ) the relation τ = (cid:80) mi =1 ∂ i holds, where ∂ i is the Dehn twist about b i . Proof.
We first note that if m = 0 we get 12 τ = 0 from a fully degeneratestar relation. Next we induct on the number of boundary curves. If m = 1this is an application of a doubly degenerate star relation. The case that m = 2 is the same with a degenerate star relation.Suppose F has m > b . . . , b m . Let b (cid:48) be theseparating curve that isolates b m and b m − from the rest of the surface. Nowby the induction step 12 τ = ∂ (cid:48) + (cid:80) m − i =1 ∂ i where ∂ (cid:48) is the Dehn twist about b (cid:48) . By the above lemma, ∂ (cid:48) = ∂ m − + ∂ m , proving the desired result. (cid:3) Given an exhaustion of S by finite type surfaces, F ⊂ F ⊂ F ⊂ · · · , as above, let P n = PMCG( F n ). We then obtain a di-rected system of groups and have lim −→ P n = PMCG c ( S ). Let f : PMCG( S ) → Z be a homomorphism. Since PMCG( S ) is a Polish group by Theorem 2.3, f is continuous, by Theorem 2.5. We obtain a continuous map f c : PMCG c ( S ) → Z , by restriction, and hence maps f n : P n → Z for each n .We claim that f n = 0 for all n . Note that then f c = 0 and sincePMCG( S ) = PMCG c ( S ), by Theorem 2.1, we will also have f = 0 andthe proof would be complete.Suppose, to the contrary, that some f n (cid:54) = 0. Since a basis for H ( P n , Z )is (cid:104) τ, ∂ , . . . , ∂ m − (cid:105) as above, we must have that f n is nonzero on some basiselement. In fact, we also have that f n is nonzero on at least one Dehn twistabout some boundary curve. Indeed, if f n ( ∂ i ) = 0 for i = 1 , . . . , m − GEORGE DOMAT AND PAUL PLUMMER a F n b i b (cid:48) i F n +1 a b i Annulus with Puncture: b i b i Pair of Pants: F n F n +1 Figure 2.
Example of the punctured annulus and pair ofpants cases. The x’s represent punctures.
OHOMOLOGY OF MCG OF GENUS ONE AND ZERO SURFACES 9 f n ( τ ) (cid:54) = 0, then using the relation that 12 τ = (cid:80) mi =1 ∂ i we must have that f n ( ∂ m ) (cid:54) = 0. Let ˆ ∂ n be a Dehn twist about a boundary curve, say ˆ b n , suchthat f n ( ˆ ∂ n ) >
0. We can assume it is positive by possibly taking an inverseDehn twist.Next we examine f k : P k → Z where k > n is the next time that an ele-ment of our exhaustion is obtained by gluing a punctured disk, a puncturedannulus, or a pair of pants to the boundary curve ˆ b n . Note that we cannothave that F k is obtained by gluing a punctured disk to ˆ b n since then ˆ ∂ n would become trivial and f k ( ˆ ∂ n ) = 0, contradicting the compatibility of the f n with our directed system.Thus we have that F k is obtained by either gluing a punctured annulus ora pair of pants to ˆ b n . Applying Lemma 3.2 we then find a new Dehn twistabout a boundary curve ˆ ∂ k such that f k ( ˆ ∂ k ) >
0. Note that we still have f k ( ˆ ∂ n ) > ∂ n i withthe property that f k ( ˆ ∂ n i ) > n i < k . Since the maps f k are acompatible sequence of maps and are compatible with f , there exists aninfinite sequence { ˆ ∂ n i } leaving every compact set such that f ( ˆ ∂ n i ) (cid:54) = 0,contradicting Lemma 2.6. (cid:3) Note that PMCG c ( S ) will have nontrivial cohomology, but it just doesnot extend (continuously) to PMCG( S ). Remark 3.4.
In this setting, PMCG( S ) does have a finite index subgroupwhich surjects onto F , the free group on two generators. This is becausewe have a surjective group homomorphism, coming from a forgetful map, toPMCG( S , ). It is a classical result that PMCG( S , ) is virtually free, asit is isomorphic to SL (2 , Z ). Taking the preimage of that finite index freesubgroup we get the desired finite index subgroup.
4. Genus zero
Let S be any surface, let X ⊂ S be a compact and totally disconnectedcollection of points in the interior of S , and let S X denote the surface ob-tained from S by marking all the points in X . Now there is a naturalhomomorphism, called the forgetful map , F : PMCG( S X ) → PMCG( S )realized by forgetting that the points in X are marked. The goal of thissection is to show that there are homomorphisms to Z from the pure map-ping class group in the genus zero case which do not factor through forgetfulmaps to finite type surfaces.Let S flute be a sphere with infinitely many isolated punctures with oneaccumulation point. We will now build a cohomology class on PMCG( S flute )which “sees” every puncture and thus cannot factor through a forgetful map.We first build a homomorphism from PMCG c ( S flute ) to Z and then showthat it extends to the closure and hence the entire pure mapping class group. GEORGE DOMAT AND PAUL PLUMMER K K γ ab γ b γ a γ a γ b K Figure 3.
Curves γ ab (in green) and γ ai and γ bi (in red andblue) on the flute surface.Let K n be a finite type exhaustion of S flute where K is a disk with twopunctures and then K n +1 is obtained from K n by gluing an annulus withone puncture to the boundary of K n .We have that the pure mapping class group of each surface K n is thepure braid group P B n +2 . H ( P B n ; Z ) has a basis coming from Dehn twistsabout each pair of punctures. The Dehn twist about the boundary curve canbe written as a product of all these Dehn twists about pairs of punctures,without inverses [FM11]. Let a and b be the two punctures in K andenumerate the other punctures of K n by 1 , . . . , n . Then the curve abouta pair of punctures is denoted γ ij where i, j > a, b and the twistabout such a curve is T γ ij . See Figure 3 for an example of these curves. Wedefine φ : PMCG( K ) → Z to be the zero map; i.e., φ sends the only basiselement, T γ ab , to 0.Next φ : PMCG( K ) → Z is defined on the basis by sending T γ ab (cid:55)→ T γ a (cid:55)→ T γ b (cid:55)→ − . Note that φ | PMCG( K ) = φ . We now define φ n +1 from φ n recursively bysetting φ n +1 | PMCG( K n ) = φ n and on the new basis elements T γ i ( n +1) (cid:55)→ i (cid:54) = a, bT γ a ( n +1) (cid:55)→ T γ b ( n +1) (cid:55)→ − . Now since the φ n are compatible we see that in the direct limit we geta map φ : PMCG c ( S flute ) → Z . We claim that φ extends to a map onPMCG c ( S flute ) = PMCG( S flute ). We first note that φ will send any Dehntwist about a curve which does not intersect K to 0. Indeed, given anycurve γ in some K n which does not intersect K we have two possibilities,either γ does not separate K and ∂K n or it does. Consider the first case. OHOMOLOGY OF MCG OF GENUS ONE AND ZERO SURFACES 11
Then γ is the boundary of two surfaces, one containing K and the other, K (cid:48) , not containing K . Thus, T γ is the product of Dehn twists about curvescontained in K (cid:48) . However, φ is zero on all of these twists since they livecompletely outside of K , so φ ( T γ ) = 0.Next suppose that γ separates K and ∂K n . Then T γ can be written inhomology as the sum of all curves about pairs of punctures in the componentof K n \ γ which contains K . The only nonzero terms in this sum come inpairs T γ ai and T γ bi and φ ( T γ ai T γ bi ) = 0. Thus we see that φ ( T γ ) = 0.To see that φ actually extends to the closure of compactly supported puremapping classes we use the following lemma which is a consequence of theproof of proposition 6.2 in [PV17]. Lemma 4.1 ([PV17]) . When S is a surface with at most one end accumu-lated by genus any f ∈ PMCG c ( S ) \ PMCG c ( S ) can be realized as an infiniteproduct of Dehn twists. Note also that any infinite product of Dehn twists can converge in thepure mapping class group only if the curves about which one twists eventu-ally leave every compact set. To check that φ extends we realize any given f ∈ PMCG c ( S flute ) \ PMCG c ( S flute ) as an infinite product of Dehn twistsabout curves which eventually leave every compact set. In particular, theyeventually have trivial intersection with K . Thus we see that φ is non-zeroon only finitely many of these Dehn twists so that φ ( f ) is finite. Further-more, if we realize f as two different infinite products then these infiniteproducts must eventually agree on every compact set so that φ ( f ) is well-defined and extends. Note that φ cannot factor through any forgetful mapto a finite type surface since forgetting any puncture would make some γ ai trivial and φ ( γ ai ) (cid:54) = 0 for all i . This gives us the following proposition. Proposition 4.2.
Let S be a surface of genus with infinitely many punc-tures. Then there exists a homomorphism from PMCG( S ) to Z that doesnot factor through a forgetful map to a sphere with finitely many punctures. Proof.
Given any such S we have mapsPMCG( S ) F −→ PMCG( S flute ) φ −→ Z where F is a forgetful map and φ is the homomorphism constructed above.Now φ ◦ F gives a homomorphism from PMCG( S ) to Z which cannot factorthrough a forgetful map to a sphere with finitely many punctures. (cid:3) In our construction of φ it was only important that φ ( T γ ai T γ bi ) = 0.Letting φ i be defined the same way as φ except set φ i ( T ai ) and φ i ( T bi ) to bezero, we get an infinite family of maps { φ i } . In fact, for any subset, A , ofpositive integers with the complement of A infinite we can define φ A to bezero at a i , b i when i ∈ A . This gives an uncountable collection of A where φ A will not factor through forgetful maps to finite type surfaces. Thus wehave GEORGE DOMAT AND PAUL PLUMMER
Corollary 4.3. If S is a genus surface with infinitely many puncturesthen there is an uncountable family of homomorphisms to Z which do notfactor through a forgetful map to a finite type surface. These two results together give Theorem 1.2.
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