Mapping class groups and moduli spaces of curves

Hain, Richard; Looijenga, Eduard

doi:10.1090/pspum/062.2/1492535

Cited by 113 publications

(143 citation statements)

References 61 publications

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“…The following result gives a partial answer to this question, and is all that we need for the calculations in this paper: a proof may be found in Section 4 of Hain-Looijenga [14]. Proposition 1.6.…”

Section: Intersection Theory On M 14mentioning

confidence: 92%

Intersection theory on \overline{ℳ}_{1,4} and elliptic Gromov-Witten invariants

Getzler

1997

J. Amer. Math. Soc.

125

111

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In this paper, we find a new relation among codimension 2 algebraic cycles in M 1,4 . The main application of the new relation is to the calculation of elliptic Gromov-Witten invariants. For example, we show that if V has no primitive cohomology in degrees above 2, the elliptic Gromov-Witten invariants are determined by the elliptic Gromov-Witten invariants, together with the rational Gromov-Witten invariants.In [12], we will prove, using mixed Hodge theory, that the cycles [M(G)], as G ranges over all stable graphs of genus 1 and valence n, span the even-dimensional homology of M 1,n , and that the new relation, together with those already known in genus 0, generate all relations among these cycles. This result is the analogue, in genus 1, of a theorem of Keel [18] in genus 0.Our new relation is closely related to a relation in A 2 (M 3 ) ⊗ Q discovered by Faber (Lemma 4.4 of [8]); the image of his relation in H 4 (M 3 , Q) under the cycle map is the same as the push-forward of our relation under the map M 1,4 → M 3 obtained by contracting the 4 tails pairwise. This suggests that our new relation should actually be a rational equivalence. 1 Let us illustrate our results with the case of the projective plane. The genus 0 and genus 1 potentials of CP 2 equalwhere t 0 , t 1 and t 2 are formal variables, of degree −2, 0 and 2 respectively, dual to the classes 1 ∈ H 0 (CP 2 , Q), ω ∈ H 2 (CP 2 , Q) and ω 2 ∈ H 4 (CP 2 , Q) respectively, and N (0) n and N (1) n are the number of rational, respectively elliptic, plane curves of degree n which meet 3n − 1, respectively 3n, generic points. Kontsevich and ManinReceived by the editors February 10, 1997 and, in revised form, June 4, 1997. 1991 Mathematics Subject Classification. Primary 14H10, 14H52, 14N10, 81T40, 81T60. Key words and phrases. Gromov-Witten invariants, moduli spaces, algebraic curves. 1 Since this paper was written, Pandharipande [26] has found a direct geometric proof of the relation of Theorem 1.8, showing that it is a linear equivalence, by means of an auxiliary moduli space of admissible covers of CP 1 .

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Section: Intersection Theory On M 14mentioning

confidence: 92%

Intersection theory on \overline{ℳ}_{1,4} and elliptic Gromov-Witten invariants

Getzler

1997

J. Amer. Math. Soc.

125

111

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“…Apparently, the reverse task of enriching mathematics with concepts known from physics had been accomplished by Witten [5]. His physical intuition had revolutionized combinatorial methods of algebraic geometry related to intersection theory on moduli space of curves [6][7][8]. In his famous paper [9] Kontsevich had provided needed mathematical justification to Witten's work.…”

Section: Motivationmentioning

confidence: 99%

“…( 1.20) This definition can be extended to describe the intersection of subvarieties V and W of a complex manifold M and it is possible to prove that Schubert calculus is just a special case of such more general algebra known in the literature as Chow algebra [8] (also as Chow ring [41]). Next, we need a notion of a divisor.…”

Section: From Grassmannians To Schubert Varietiesmentioning

confidence: 99%

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Kholodenko¹

2002

Journal of Geometry and Physics

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In previous publications ( J. Geom. Phys.38 (2001) 81-139 and references therein ) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface.With help of this function the dynamical transition from pseudo-Anosov to periodic (Seifert-fibered) regime was studied. In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich (Comm.Math.Phys. 147 (1992) 1-23 ) inspired by earlier work of Witten on topological two dimensional quantum gravity.To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some non traditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics.

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“…Faber [15] made a beautiful conjecture about the structure of R * (M g ). There have been done many works related to and inspired by Faber's conjecture (we refer to survey papers [30][53] [105] for some of the recent results including enhancements of Faber's original conjecture). However the most difficult part of Faber's conjecture, which claims that R * (M g ) should be a Poincaré duality algebra of dimension 2g − 4, remains unsettled.…”

Section: Tautological Algebra Of the Mapping Class Groupmentioning

confidence: 99%

Cohomological structure of the mapping class group and beyond

Morita¹

2006

Proceedings of Symposia in Pure Mathematics

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Abstract. In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces.Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders.

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Mapping class groups and moduli spaces of curves

Cited by 113 publications

References 61 publications

Intersection theory on \overline{ℳ}_{1,4} and elliptic Gromov-Witten invariants

Intersection theory on \overline{ℳ}_{1,4} and elliptic Gromov-Witten invariants

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Cohomological structure of the mapping class group and beyond

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