The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces (Q2848313)

Denote by \(M_g\) the moduli space of compact Riemann surfaces (or smooth projective algebraic curves) of genus \(g\geq 2\). Thirty years ago, in his seminal paper [Towards an enumerative geometry of the moduli space of curves. Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271--328 (1983; Zbl 0554.14008)], \textit{D. Mumford} initiated the systematic study of both the Chow ring and the rational cohomology ring of \(M_g\) in the course of which he formulated a conjecture stating that the rational cohomology ring \(H^*(M_g;\mathbb{Q})\) is a polynomial \(\mathbb{Q}\)-algebra generated by certain classes \(\kappa_i\) of dimension \(2i\), \(i\geq 1\) the so-called Mumford-Miller-Morita classes. This has since been known as the ``Mumford conjecture'', and has become a theorem in 2007, due to its proof by \textit{I. Madsen} and \textit{M. Weiss} [Ann. Math. (2) 165, No. 3, 843--941 (2007; Zbl 1156.14021)]. Another basic ingredient, in the proof of Mumford's conjecture, is a stability theorem by \textit{J. L. Harer} [Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)], which implies that the rational cohomology of \(M_g\) is independent of the genus \(g\) in a range of dimensions.NEWLINENEWLINE Actually, the theorems by Harer and Madsen-Weiss are both results concerning the integral homology of the mapping class groups of closed, smooth, oriented surfaces of genus \(g\) and their relations to the algebraic statement of Mumford's conjecture are indeed a fairly subtle matter.NEWLINENEWLINE The paper under review consists of four lectures delivered at the IAS/Park City Mathematics Institute in July 2011. The aim of these lectures is to explain just these connections between Mumford's conjecture, the Madsen-Weiss theorem, and Harer's stabilty theorem to graduate students and non-specialists interested in these topics. In Lecture 1, the author gives a brief introduction to moduli spaces of Riemann surfaces, mapping class groups and their classifying spaces, the Mumford-Morita-Miller classes as characteristic classes for certain surface bundles, and first statements of the theorems of Harer and Madsen-Weiss, respectively. Lecture 2 describes a general strategy for proving homological stability results for families of groups, with particular emphasis on the case of mapping class groups of surfaces. One of the main references, in this context, is the author's recent survey article [Homological stability for mapping class groups of surfaces, in: Handbook of moduli (G. Parkas, ed.) (2012)]. Lecture 3 presents a spectral sequence argument due to \textit{O. Randal-Williams} [Resolutions of moduli spaces and homological stability, Preprint, \url{arxiv:0909.4278}], which will be used in the sequel to prove homological stability for the mapping class groups of surfaces. This is done in Lecture 4, after a last ingredient of the proof is sketched. This ingredient concerns a connectivity property of the occurring arc complexes for surfaces with boundary.NEWLINENEWLINE As for a proof of the Madsen-Weiss theorem, this series of lectures is supplemented by the subsequent four lectures given by \textit{S. Galatius} [in: Moduli spaces of Riemann surfaces. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Study (IAS). IAS/Park City Mathematics Series 20, 139--167 (2013; Zbl 1281.14023)].NEWLINENEWLINEFor the entire collection see [Zbl 1272.30002].