A Very Simple Estimate Of Rational Homological Dimension Of Moduli Spaces Of Riemann Surfaces With Boundary And Marked Points
aa r X i v : . [ m a t h . QA ] M a r A VERY SIMPLE ESTIMATE OF THE RATIONALHOMOLOGICAL DIMENSION OF MODULI SPACES OFRIEMANN SURFACES WITH BOUNDARY AND MARKEDPOINTS
HAO YU
Abstract.
The moduli spaces of compact and connected Riemann surfaceshas been a central topic in modern mathematics. Hence their homological di-mensions become important invariants. Motivated by the emergence of open-closed string theory and its mathematical counterparts, we give a coarse es-timate of rational homological dimension of Riemann surfaces with possibleboundary and marked points(can lie on both interior and boundary). Wehope it will have applications in open-closed theory, for example, open-closedGromov-Witten theory in the future. Introduction
The moduli spaces of compact and connected Riemann surfaces with markedpoints has been a central and active topic in topology, differential geometry, alge-braic geometry and mathematical physics. The estimate of its homological dimen-sion, i.e., the greatest degree for which the homology group over a coefficient ring R , typically Z or Q , doesn’t vanish, is thus a fundamental interest. The estimateof relevant quantities along this line has been worked out before (for example see[Har86, M17]). There are several applications of this estimate in the literature. Forexample, in [Cos07] Costello gives a uniqueness of the solutions of Quantum masterequation in topological conformal field theory by using a coarse estimate of thishomological dimension. In this paper, we give an analogue estimate of the rationalhomological dimensions of the moduli spaces of Riemann surfaces with (ordered orunordered) boundaries and with (ordered or unordered) punctures/marked pointson the interior and boundaries of the surfaces. The latter is known as open-closedcased in mathematical physics or algebraic geometry literature. We refer the readers[Liu02] for the thorough study of these spaces. Although our estimate is still coarse,analogue to the works mentioned above, we hope it will be useful in open-closedtheory, e.g., open-closed Gromove-Witten theory, or open-closed string theory.In the following sections, we actually give two estimates of the homological di-mensions of moduli spaces of Riemann surfaces with boundary and punctures/markedpoints respectively. It turns out that the latter is stronger than the former. How-ever, the method used in the deduction of the first estimate is fundamental althoughstill subtle, involving tools in both geometry(Teichmuller spaces and classifyingspaces) and algebra(mapping class groups and spectral sequences), as well as thededicate analysis of special cases. It has independent interests, and we believe itwill be useful on its own. The method for the second estimate uses the establisheddeep theory of virtual duality groups, so it will be a little simpler than the first. From hyperbolic geometry and the theory of Riemann surfaces, we know that aRiemann surface with r boundary components and n punctures is equivalent tothe surface equipped with a complete, finite area hyperbolic metric with geodesicboundary (to see this, double the surface, and we have a unique hyperbolic geo-metric structure corresponding to the complex structure of the double, and theboundary is invariant under the involution, so it is geodesic). Let H S be the spaceof such metrics on the surface S , let Dif f ( S ) denote the group of oriented preserv-ing diffeomorphisms of S which fixes each puncture and boundary component, and Dif f S be the subgroup consisting of those diffeomorphisms which are isotropic viasuch diffeomorphisms to the identity. And M od ( S ) = Dif f ( S ) /Dif f S is knownas Mapping class group of S . Dif f ( S ) and Dif f S acts on H S (via metric pullback) and the quotient space T S = H S /Dif f S is called Teichmuller space of S .The space M S = H S /Dif f ( S ) is the Moduli space of S , whose points parameter-ize the isomorphism classes of complete, finite area hyperbolic metrics on S withordered punctures and boundary components. From the definition, we also know T S /M od ( S ) = M S .In the remaining of this paper, we will denote S b,~mg,n to be a surface of genus g with b boundary components, labelled 1 , , . . . , b and n interior punctures/markedpoints, ~m = ( m , m , ..., m b ) punctures/marked points on the boundary, with m i punctures/marked points on the i -th boundary component. And let m = P bi =1 m i .If ~m = ~
0, we denote it as S bg,n , and furthermore if b = 0 or n = 0 or both are 0 wedenote it by S g,n , S bg or S g , respectively.2. Estimate I
We will prove in this section the following theorem.
Theorem 2.0.1.
Let M b,~mg,n be moduli space of Riemann surfaces of genus g with b boundary components, labelled , , . . . , b and n interior punctures which is alsolabelled, ~m = ( m , m , · · · , m b ) punctures on the boundary, with m i punctures onthe i -th boundary component, m = P bi =1 m i , then we have the following estimatefor the homology groups: (1) H i ( M S b,~mg,n , Q ) = 0 for i ≥ g − n + 3 b + m except ( g, n, b, ~m ) = (0 , , , , (0 , , , , (0 , , , , (0 , , , ((1 , , (0 , , , (1 , , (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , . For oriented surfaces of genus g (with or without boundaries and punctures),it is well known that the corresponding Teichmuller space is contractible (it ishomeomorphism to a Eucliean space). Specifically, if S is an oriented surface ofgenus g with b boundary components and n punctures in the interior, then [FM08]. T S = R g − n +3 b . And we know ([FM08]) that the mapping class group M od ( S )acts properly discontinuously on Teichmuller space T S , and the stabilizer is finitefor each point, so by the Borel construction E := T S × Mod ( S ) EM od ( S ), EM od ( S )is a universal principle M od ( S ) bundle and BM od ( S ) = EM od ( S ) /M od ( S ) is the classifying space of M od ( S ) (which classifies all principle M od ( S ) bundles on anytopological space). The M od ( S ) acts via diagonal action. Since T S is contractible, E is homotopic to BM od ( S ), and the projection E → T S /M od ( S ) = M S is afibration with fiber EM od ( S ) /stab ( x ) = Bstab ( x ). Since stab ( x ) is a finite group, VERY SIMPLE ESTIMATE OF THE RATIONAL HOMOLOGICAL DIMENSION OF MODULI SPACES OF RIEMANN SURFACES WITH BOUNDARY AND MARKED POINTS3 the cohomology of the fiber vanishes over Q . That is, we have H ∗ ( M od ( S ) , Q ) = H ∗ ( BM od ( S ) , Q ) = H ∗ ( M S, Q ).In the rest of the paper, we will consider the surfaces with possibly punc-tures/marked points on the boundary.For mapping class groups, we have the Birman exact sequence:(2) 1 → π ( S bg,n − ) → M od ( S bg,n ) → M od ( S bg,n − ) → g = 0 , b + n ≤ , g = 1 , b + n ≤ → π ( C ( S bg,n − , k )) → M od ( S bg,n + k − ) → M od ( S bg,n − ) → S , C ( S, k ) is the configuration space of k distinct,orderedpoints in S . That is, C ( S, k ) = S k − BigDiag ( S k )where S k is the k -fold cartesian product of S and BigDiag( S k ) is the Big Diag of S k , that is, the subset of S k with at least two coordinates being equal. A variantof it is(4) 1 → π ( U T S bg,n ) → M od ( S bg,n ) → M S b − n → U T S b − g,n is the unit tangent bundle (spherized tangent bundle) of S b − g,n . Thespecial cases above are also excluded. Those sequences are important tools for ourcalculation of mapping class groups. We will derive a sequence similar in spirit, formapping class groups of surfaces with punctures/marked points on the boundary.More precisely, assume S is an oriented surface of genus g with b boundary com-ponents and n punctures/marked points in the interior and m = ( m , m , . . . , m b )punctures/marked points on the boundary, with m i punctures/marked points onthe i -th boundary component. M od ( S ) is the mapping class group, i.e., the group ofequivalence classes of oriented-preserving homeomorphisms of S which fixes bound-ary component set-wise and fixes each punctures individually, the equivalence rela-tion is given by isotropy of the same type between them.The following fact is obvious.(5) M od ( S b, ( m ,m ,...,m i +1 ,...,m b ) g,n ) ∼ = M od ( S b, ( m ,m ,...,m i ,...,m b ) g,n ) if m i > ~m =( m , m , . . . , m b ) punctures/marked points to that with each m i at most 1. Forthis kind of surfaces we have the following short exact sequence:(6) ( Z ) r → M od ( S b,~m ) g,n ) → M od ( S bg,n ) if χ ( S b,ng ) < r is the number of i for which m i = 0, and m i = 0 or 1 for i = 1 , , . . . , b .In order to achieve the desired homological dimension estimate, we need thefollowing two key ingredients:(i) The short exact sequence of groups corresponding to the fibration sequenceof classifying spaces.If G , G , G are groups,(7) 1 → G → G → G → BG → BG → BG is a fibration, where BG i are classifying spaces of G i .This is the standard fact in classifying space theory.(ii) If F → E → B is a Serre fibration, H i ( B, Q ) = 0 , H j ( F, Q ) = 0 when i ≥ n, j ≥ m + 1, and H ∗ ( B, Q ) is of finite rank, then H k ( E, Q ) = 0 if k ≥ n + m . In fact, by the Serre spectral sequence,(9) E p,q = H p ( F, H q ( B, Q )) ⇒ H p + q ( E, Q )the homology group we are going to compute is of index p + q = k ≥ n + m ,so either p ≥ n or q ≥ m + 1. Using the assumption of homology groups H i ( B, Q ), H j ( F, Q ) and the rank finiteness of H ∗ ( B, Q ), we immediately getthe conclusion.With all the preparation above, we can now prove the theorem 2.0.1.
Proof.
As before, S b,~mg,n is a surface with possibly punctures/marked points on theinterior and boundary.(1) there are some i such that m i ≥
2. In this case, we can use (5) to reduceit to the case that m i = 0, or 1 for all i , i.e., M od ( S b,~mg,n ) ∼ = M od ( S b, ~m ′ g,n ) ,where m i = 1, if m i ≥ m i = 0 or 1. In this case, we use the short exact sequence (6),provided that the Euler class of punctures ”filled in” surface is negative,and the induced fibration of classifying spaces mentioned before. Since theclassifying space of group Z r is ( S ) r ( i.e, K ( Z r , S ) r → M S b,~mg,n → M S bg,n Since ( S ) r is an r dimensional manifold, we have H i (( S ) r , Q ) = 0 if i ≥ r + 1. Using (6), we have an induction:if H i ( M S bg,n , Q ) = 0, for i ≥ g − n + 3 b , then H j ( M S b,~mg,n , Q ) = 0 for j ≥ g − n + 3 b + r , where the condition χ ( S bg,n ) < π ( U T S b − g,n ) is U T S b − g,n (since U T S b − g,n is K ( π , U T S b − g,n → M S bg,n → M S b − g,n Because
U T S b − g,n is 3 dimensional manifold, H i ( U T S b − g,n , Q ) = 0 , for i ≥ H i ( M S b − g,n , Q ) = 0, for i ≥ g − n + 3( b − H j ( M S bg,n , Q ) = 0,for j ≥ g − n + 3 b , where the condition χ ( S b − g,n ) < b = 0. In this case, in [Cos07], it has established that H i ( M S g,n , Q ) = 0,for i ≥ g − n except ( g, n ) = (0 , S b,~mg,n has χ ( S b,~mg,n ) < S b,~mg,n . Then using the induction (asabove) we conclude that H i ( M S b,~mg,n , Q ) = 0 for i ≥ g − n +3 b , except ( g, n, b ) =(1 , , , (1 , , , (1 , , , (0 , , , (0 , , , (0 , , , (0 , , , (0 , , , (0 , , , (0 , , , (0 , , , (0 , , , (0 , , VERY SIMPLE ESTIMATE OF THE RATIONAL HOMOLOGICAL DIMENSION OF MODULI SPACES OF RIEMANN SURFACES WITH BOUNDARY AND MARKED POINTS5 have ( g, n, b ) = (1 , , , (1 , , , (0 , , , (0 , , , (0 , , , (0 , , χ ( S bg,n ) = 2 − g − ( b + n ) <
0, this includes all the cases except ( g, n, b ) =(0 , , , (0 , , , (0 , , b >
0. So if ( g, n, b ) is not those numbers, ac-cording to (i), we can reduce the surface with punctures on the boundary to the onewithout punctures on the boundary by (i), except those bad cases. Other surfaces allsatisfy the homological dimension estimates. Then by induction, the original surfacesatisfies the homological dimension estimate. So we are left the cases ( g, n, b, ~m ) =(0 , , , ~m ) , (0 , , , ~m ) , (0 , , , ~m ) , (1 , , , ~m ) , (0 , , , ~m ) , (0 , , , ~m ),and (0 , , , ~m ).We need to treat them individually.Lets start from g = 1.If ( g, n, b, ~m ) = (1 , , , ~m ), then since the dimension of the moduli space M S , is 3 (it is the complement of trileaf knot) and H ( M S , , Q ) = H ( M S , , Q ) = 0(see [MK02]), so in this case the homological dimension estimate is satisfied.For g = 0, if ( g, n, b, ~m ) = (0 , , , ~m ), then since the moduli space of S , , i.e.,a pair of pants, is the same as the Teichmuller space of S , , for which T S , = R , so the homological dimension estimate is obviously satisfied. Then for all ~m , thehomological dimension is satisfied. The same is true for ( g, n, b, ~m ) = (0 , , , ~m ).if ( g, n, b, ~m ) = (0 , , , ~m ), then since M S , = R , H i ( M S , , Q ) = 0 when i ≥
1, sowhen m ≥
2, the homological dimension estimate is satisfied. The only cases left is(0 , , , , (0 , , , g, n, b, ~m ) = (0 , , , ~m ), then the condition χ ( S ,~m , ) < m ≥
3, and thedimension of the moduli space of S , , is 0, so when m >
4, the homological dimen-sion estimate is satisfied. This leaves the cases ( g, n, b, ~m ) = (0 , , , , (0 , , , g, n, b, ~m ) = (0 , , , ~m ), then the moduli space of S , is an interval (0 , H i ( M S , , , , Q ) = 0 when i ≥
1. Andthe moduli space of S , (1 , , is of dimension 2 and homotopic to S , so the homol-ogy group H i ( M S , (1 , , , Q ) = 0 when i ≥
2. Thus when m = (0 , , (1 , , (1 , g, n, b, ~m ) =(0 , , , (1 , , (0 , , , (1 , g, n, b, ~m ) = (0 , , , ~m ), because m ≥ M S , , is 0, we have that when m ≥
3, then the homological dimension estimateis satisfied. This leaves ( g, n, b, ~m ) = (0 , , , , (0 , , , HAO YU satisfy the homological dimension estimate).If there are no punctures on the boundary, then when ( g, n, b ) = (1 , , , , , (0 , , , (0 , , g, n, b ) = (0 , , , (0 , , H i ( M S b,~mg,n , Q ) = 0, for i ≥ g − n + 3 b + m , except ( g, n, b, ~m ) =(0 , , , , (0 , , , , (0 , , , , (0 , , , (1 , , (0 , , , (1 , , (0 , , , , (0 , , , , (0 , , , , (0 , , , Remark . In fact, it is already from the above analysis that the homologicaldimension estimates can be improved to H i ( M S b,~mg,n , Q ) = 0 when i ≥ g − n +3 b + q , q is the number of nonzero entries in ~m . (cid:3) Estimate II
In this section, we will get another homological dimension estimate for modulispace over coefficient Q , which is stronger than the one we have proposed above.Denote d ( g, n, b ) = g − , n = b = 04 g + 2 b + n − , g > , n + b > b + n − , otherwiseand q is the number of nonzero entries in ~m . Theorem 3.0.1. H i ( M S b,~mg,n , Q ) = 0 when i > d ( g, n, b ) + q .Proof. (1) It is easy to see that the moduli space M S b,~mg,n is homotopy equivalent to M S b,~mg,n , where m i = 0 or 1, depending on whether m i = 0 or m i ≥
0. And M S b,~mg,n is a fibration with fiber homeomorphic to ( S ) p over moduli space M S bg,n , where p is the number of non-zero entries in ~m . (2) In [Har86], it is shown that when χ ( S bg,n ) < M od ( S bg,n ) is a virtual duality group of dimension d ( g, n, b ),in particular, the virtual cohomological dimension of M od ( S bg,n ) is d ( g, n, b ). Sincefor a duality group the cohomological dimension and homological dimension agrees([BE73]), we have H i ( M S bg,n , Q ) = 0, if i > d ( g, n, b ). Summarizing the above two,and using the Serre spectral sequence and (ii) in the proof of theorem (2.0.1), weget the conclusion. We still have some cases which need to be treated separately,they are ( g, n, b, ~m ) = (0 , , , ~m ) , (0 , , , ~m ) , (0 , , , ~m ).When ( g, n, b, ~m ) = (0 , , , ~m ) we need only consider m >
0. In this case, since H ∗ ( M S b,~mg,n , Q ) = H ∗ ( M S b, (1) g,n , Q ) and S b, (1) g,n is of dimension 0, so H i ( M S ,~m , , Q ) = 0when i ≥
1. Since d (0 , ,
1) = 0 , d (0 , ,
1) + q = 1, the result is stronger.When ( g, n, b, ~m ) = (0 , , , ~m ), ~m must has at least one nonzero entry. Fromthe result in the proof of theorem 1, we have H i ( M S , ( m, , , Q ) = 0 when i ≥
1, itis stronger. And we have H i ( M S , ( m ,m , , Q ) = 0 (i.e., q = 2) when i ≥
2, which is
VERY SIMPLE ESTIMATE OF THE RATIONAL HOMOLOGICAL DIMENSION OF MODULI SPACES OF RIEMANN SURFACES WITH BOUNDARY AND MARKED POINTS7 also stronger.When ( g, n, b, ~m ) = (0 , , , ~m ), then we have m ≥
3. When m = 3 , S ,~m , is ofdimension 0, so H i ( M S ,~m , , Q ) = 0 when i ≥
1, which is the same as the aboveestimate. (cid:3)
In this paper, we further discuss moduli spaces of surfaces with unordered punc-tures and boundary components, which is quite useful when the boundary andmarked points are equipped with group actions of permutations. We similarly let˜ S b,~mg,n be a surface with unordered punctures and boundary component. Theorem 3.0.2. H i ( M ˜ S b,~mg,n , Q ) = 0 when i > d ( g, n, b ) + q This can be shown as follows.
Proof.
It is an exact sequence(12) 0 → M od ( S b,~mg,n ) → M od ( ˜ S b,~mg,n ) → S n × S b ⋉ ( Z m × · · · × Z m b ) → M od ( S b,~mg,n ) → M od ( ˜ S b,~mg,n ) is the canonical one, and M od ( ˜ S b,~mg,n ) → S n × S b ⋉ ( Z m × · · · × Z m b ) records how the homeomorphism permutes punctures.The third nonzero term of the exact sequence being a finite group implies thatif the homology group over Q of the second nonzero term is 0 then so it is for thefirst nonzero term. This can be deduced from the transfer map for group homologyassociated to a group G and its finite-index subgroup H (see [Brown82],[AJ04]).The composition of the restriction map with the transfer map is: T r GH ◦ Res GH ( x ) =[ G : H ] x , for x ∈ H ∗ ( G, Q ). Because H ∗ ( G, Q ) is a Q vector space, it means thatthe restriction map, Res GH : H ∗ ( G, Q ) → H ∗ ( H, Q ), is injective. (cid:3)
References [AJ04] A. Adem, R. James Milgram,
Cohomology of Finite Groups , (2004), Grundlehren derMathematischen Wissenschaften, 309, Springer-Verlag.[BE73] R.Bieri, B.Eckmann,
Groups with Homological Duality Generalizing Poincare Duality ,Inventiones Math. 20, 103-124 (1973).[Brown82] Kenneth.S.Brown,
Cohomology of Groups , (1982), Graduate Text in Mathematics, 87,Springer Verlag.[Cos07] K. Costello,
The Gromov-Witten potential associated to a TCFT (2007), math.QA/0509264.[FM08] B. Farb and D. Margalit,
A primper on mapping class group
On the cohomological dimension of the moduli space of Riemann sur-faces , Duke Mathematcal Journal, 2017, 1463-1515.[Har86] J. L. Harer,
The virtual cohomological dimension of the mapping class group of an ori-entable surface , Invent. Math. 84(1), 157C176 (1986).[Liu02] C.-C. M. Liu,
Moduli of J-holomorphic curves with Lagrangian boundary conditions andopen Gromov-Witten invariants for an S1-equivariant pair , (2002), math.SG/0210257.[MK02] M. Korkmaz,
Low-dimensional homology groups of mapping class groups: a survey , TurkJ Math 26 (2002), 101-114.
Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA
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