# The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

@article{Etingof2005TheCR, title={The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points}, author={Pavel Etingof and Andr{\'e} Henriques and Joel Kamnitzer and Eric M. Rains}, journal={Annals of Mathematics}, year={2005}, volume={171}, pages={731-777} }

We compute the Poincare polynomial and the cohomology algebra with rational coefficients of the manifold M_n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of M_n. As was shown by the fourth author, the cohomology of M_n does not have odd torsion, so that the above determines the additive structure of the integral homology…

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## References

SHOWING 1-10 OF 40 REFERENCES

### Chen Lie algebras

- Mathematics
- 2004

The Chen groups of a finitely presented group G are the lower central series quotients of its maximal metabelian quotient G/G″. The direct sum of the Chen groups is a graded Lie algebra, with bracket…

### On lie k-algebras

- Mathematics
- 1995

Abstract We define the notion of a "Lie k -algebra" to be a ( k + 1)-ary skew-symmetric operation on a bigraded vector space which satisfies a certain relation of degree 2 k + 1. The notion of Lie…

### Koszul duality for Operads

- Mathematics
- 1994

(0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] and…

### Quantum cohomology of a product (with Appendix by R. Kaufmann2)

- Mathematics
- 1996

The operation of tensor product of Cohomological Field Theories (or algebras over genus zero moduli operad) introduced in an earlier paper by the authors is described in full detail, and the proof of…

### THE ACTION OF Sn ON THE COHOMOLOGY OF M0,n(R)

- Mathematics
- 2006

In recent work by Etingof, Henriques, Kamnitzer, and the author, a presentation and explicit basis was given for the rational co-homology of the real locus M0,n(R) of the moduli space of stable genus…

### Conjugation spaces

- Mathematics
- 2004

There are classical examples of spaces X with an involution τ whose mod2-comhomology ring resembles that of their ﬁxed point set X τ : there is a ring isomorphism κ : H 2 ∗ ( X ) ≈ H ∗ ( X τ ). Such…

### Non-trivial 1-classes in the homology of the real moduli spaces M-bar_{0,n} and related structures

- Mathematics
- 2002

We give lower bounds for the rank of the first homology group of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow…

### Gromov-Witten classes, quantum cohomology, and enumerative geometry

- Mathematics
- 1994

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic…