Mapping class groups of low genus and their cohomology

Symplectic 4 manifolds, stein domains, seiberg witten. Newest mappingclassgroup questions mathematics stack. This embedding induces a homology isomorphism in degrees. The first cohomology group is the quotient of the socalled crossed homomorphisms, i. There are intriguing analogies between automorphism groups of. Homology of hyperelliptic mapping class groups for.

Jun 17, 2019 we prove a new kind of stabilisation result, secondary homological stability, for the homology of mapping class groups of orientable surfaces with one boundary component. Simple examples of mapping class groups and basic properties of dehn twists will also be given. So in general, there is a short exact sequence of groups. The ordinary cohomology with rational coefficients of the delooping of the stable mapping class group of 2dimensional manifolds hence essentially the orbifold cohomology of the moduli stack of complex curves is the content of mumford. Nov 16, 20 in this paper we apply the theory of finitely generated fimodules developed by church et al. More precisely, the braid group on n strands is naturally isomorphic to the mapping class group of a disc with n punctures the dehnnielsenbaer theorem.

I havent read benson and cohen, mapping class groups of low genus and their cohomology, but the mathematical. Cohomological topics in group theory falvey memorial library. Combining our result with recent work of madsen and weiss, we obtain that the classifying space of the stable mapping. Frederick r cohen this series of papers is aimed towards the calculation of the cohomology of the mapping class group of a closed oriented surface of genus two. Our results together with those from apv17 give an almost complete picture of the rst integral cohomology of these big pure mapping class groups. Ams transactions of the american mathematical society. Homology of hyperelliptic mapping class groups for surfaces. For example, although the first homology group of the mapping class group in genus one. The only piece still missing is an explicit description of the cohomology in the genus zero case.

We prove that certain obstructions to the existence of a faithful linear representation do not exist in mg. Problems on mapping class groups and related topics. The center of some braid groups and the farrell cohomology. D j benson, f r cohen, mapping class groups of low genus and their cohomology, mem. We also construct some subgroups of finite index in the mapping class group of a genus 3 surface and calculate their first cohomology groups, which all turn out. In his list we find only three prime factors in the orders of the groups. For example, any finite group can be realized as the mapping class group and also the isometry group of a compact hyperbolic 3manifold. In particular, we collect calculations of the cohomology groups for the mapping class groups of lowgenus orientable and nonorientable. But, unfortunately, their method do not apply to higher genus. Mapping class groups of low genus and their cohomology material type. In particular, the mapping class group of a surface is a finitely generated group. Homology of the mapping class group 2,1 for surfaces of. On the cohomology of pure mapping class groups as fimodules.

X y determines a homomorphism from the cohomology ring of y to that of x. I found many results on the rational cohomology in particular of the stable mcg, but no results on the integral cohomology or even cohomology with the above coefficients in degree three. For example, although the rst homology group of the mapping class group in genus one case is known, it seems that it does not appear in the literature. Braid groups can be defined as the mapping class groups of a disc with punctures. Twenty years ago, mumford initiated the systematic study of the cohomology ring of moduli spaces of riemann surfaces.

Hilden, on the mapping class groups of closed surfaces as covering spaces, in. Low dimensional homology groups of mapping class groups. There is no new result in the paper, but there are some new proofs. Homological stability for the mapping class groups of nonorientable surfaces nathalie wahl abstract. The kazhdan property of the mapping class group of. As an application, we use these to show that the primary part. Connections with 3manifolds, symplectic geometry and algebraic geometry 129 chapter 9. We identify the quotient group with the mapping class group of a surface with a marked point or a surface with one parametrized boundary component. Mapping class groups of low genus and their cohomology creador. We identify the quotient group with the mapping class group of a surface with a marked point or a surface with. We explore aspects of these analogies, focusing on.

On the cohomology of pure mapping class groups as fi. Mapping class groups of low genus and their cohomology. Cohen, mapping class groups of low genus and their cohomology, mem. The purpose of this article is to compute the mod 2 cohomology of. In the lowdimensional topology literature, the mapping class group of x is usually.

Mapping class groups of low genus and their cohomology 0. The second homology groups of mapping class groups of. The first statement is due to, see also at sphere eversion. Sp2g,z on some graded modules associated with the lower as well as other. If is closed and is a homeomorphism of then we can define an automorphism. Sp 2g,z on some graded modules associated with the lower as well as other. We also construct some subgroups of finite index in the mapping class group of a genus 3 surface and calculate their first cohomology groups, which all. Strings and the stable cohomology of mapping class. Mapping class groups of low genus and their cohomology book. Since each component is contractible, there are natural isomorphisms of integral cohomology groups. The mod 2 cohomology of the mapping class group of the klein.

We show that the mapping class group of a closed surface of genus 2 does not satisfy the kazhdan property by constructing subgroups of finite index having a nonvanishing first cohomology group. Mapping class groups are also called homeotopy groups in the literature. Abstract the torelli group and representations of mapping class groups tara e. I am currently studying mapping class groups, and would like to know why one uses orientationpreserving homeomorphisms in their definition. Abstract in this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. Some known results on higher cohomology are also mentioned. In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. In mathematics, in the subfield of geometric topology, the mapping class group is an important. Nevertheless, few explicit computations are known for the other examples in table 1.

We prove a new kind of stabilisation result, secondary homological stability, for the homology of mapping class groups of orientable surfaces with one boundary component. Using the existence of certain symplectic submanifolds in symplectic 4manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal lefschetz fibrations over surfaces of positive genus. The mod 2 cohomology of the mapping class group of the. Automorphism groups of free groups, outer space, group cohomology. Cohenmapping class groups of low genus and their cohomology. Pdf farrell cohomology of low genus pure mapping class. Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be. Presented at the 1993 asme winter annual meeting, new orleans, louisiana, november 28december 3, 1993, p. In the second map, the group g,s is the mapping class group of a surface of genus g with s0 punctures, which may be permuted. Continuous bounded cohomology of locally compact groups by.

Mapping class groups of 3manifolds have received considerable study as well, and are closely related to mapping class groups of 2manifolds. This follows from the definition of cochains above. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. Similarly, the subgroup that acts as the identity on all the homology groups of m is. The torelli group and representations of mapping class groups. The second and third are due to earleeells 67, gramain 73. Linear representation stable bounds for the integral. Around the same time, harer proved that the homology of the mapping class groups of oriented surfaces is independent of the genus in low degrees, increasing with the genus. Bounded cohomology and nonuniform perfection of mapping. The theorem may be stated in a simplier form in terms of the notion of mapping class groups, which we will discuss later. The cohomology of automorphism groups of free groups. The torelli group and representations of mapping class groups tara e. The second homology groups of mapping class groups of orientable surfaces article in mathematical proceedings of the cambridge philosophical society 403.

Lowdimensional homology groups of mapping class groups. We prove that the homology of the mapping class groups of nonorientable surfaces stabilizes with the genus of the surface. Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. Ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Mapping class group factorizations and symplectic 4manifolds. The least possible cardinality of dehn twists generating the mapping class group of a closed surface of genus g. The cohomology of automorphism groups of free groups karenvogtmann. There is a vast literature on the cohomology of configuration spaces, with specific computations in special cases contained, for example, in 1, 10, 15.

We explore aspects of these analogies, focusing on cohomological. Mapping class groups of low genus and their cohomology ebook. The cohomology of automorphism groups of free groups karen vogtmann. Strings and the stable cohomology of mapping class groups.