A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves

Goulden, I.; Harer, J.; Jackson, D.

doi:10.1090/S0002-9947-01-02876-8

A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves
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by I. P. Goulden, J. L. Harer and D. M. Jackson PDF

Trans. Amer. Math. Soc. 353 (2001), 4405-4427 Request permission

Abstract:

We determine an expression $\xi ^s_g(\gamma )$ for the virtual Euler characteristics of the moduli spaces of $s$-pointed real $(\gamma =1/2$) and complex ($\gamma =1$) algebraic curves. In particular, for the space of real curves of genus $g$ with a fixed point free involution, we find that the Euler characteristic is $(-2)^{s-1}(1-2^{g-1})(g+s-2)!B_g/g!$ where $B_g$ is the $g$th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is $(-1)^{s}(g+s-2)!B_{g+1}/(g+1)(g-1)!$

The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter $\gamma$ that permits specialization of the formula to the real and complex cases. This suggests that $\xi ^s_g(\gamma )$ itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.

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