The stable moduli space of Riemann surfaces: Mumford's conjecture

By I. Madsen, Michael Weiss. 2002

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D.Mumford conjectured in (30) that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes i of di- mension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by B 1, where 1 is the group of isotopy classes of automorphisms of a smooth oriented connected surface of "large" genus. Tillmann's insight (41) that the plus construction makes B 1 into an infinite loop space led to a stable homo- topy version of Mumford's conjecture, stronger than the original (22). We prove the stronger version, relying on Harer's stability theorem (15), Vassiliev's theorem concerning spaces of functions with moderate singularities (43), (42) and methods from homotopy theory.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c21ea10ee2c73a0c61a5204dfcc51756037fac98

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Citations in the corpus

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields (2009, neutral): “(1) The configuration space Confn of n points in the plane; (2) The moduli spaceMg of smooth projective curves of genus g [26], [31]; (3) Classifying spaces of arithmetic groups, e.”
Stable homology of automorphism groups of free groups (2006, methodology): “1) is the analogue of Madsen-Weiss’ generalized Mumford conjecture ([MW07]; see also [GMTW09]).”
Resolutions of moduli spaces and homological stability (2009, background): “By the affirmed Mumford conjecture [20, 16], there are no relations between the κi in H (Γ∞;Q) so neither are there any in H(Γg,1;Q) in degrees ∗ ≤ ⌊2g/3⌋.”
Representation theory and homological stability (2010, background): “cture). The Sp2g Z–invariant stable rational cohomology of Ig,1 is generated as a Q–algebra by {e2,e4,e6,...}. Note that all the ei generate the stable rational cohomology of Modg,1, by Madsen– Weiss [MW], and the odd classes e1,e3,e5,...vanish when restricted to Ig,1. Morita’s Conjecture predicts t”
Relations in the tautological ring of the moduli space of curves (2013, neutral): “The first is Mumford’s conjecture, proven in 2002 by Madsen and Weiss [11], lim g→∞ H(Mg,Q) = Q[κ1, κ2, κ3, .”
Topological methods in moduli theory (2014, neutral): “The ring structure of the cohomology of the “stable mapping class group” is described by a conjecture of Mumford [307], that has been proven by Madsen and Weiss [274].”
Monoids of moduli spaces of manifolds (2009, methodology): “Corollary C (Madsen–Weiss [10]) There is a homology equivalence”
On the Classification of Topological Field Theories (2009, background): “1 was proven by Madsen and Weiss in [19].”
A vanishing theorem for characteristic classes of odd-dimensional manifold bundles (2009, background): “The Madsen-Tillmann-Weiss map (or MTW-map) of the bundle f : E → B is the composition (2.4.2) αE := κE ◦ PTf : Σ∞B+ → MTSO(n), which is defined uniquely up to a contractible space of choices.”
Stabilization for mapping class groups of 3-manifolds (2007, methodology): “Stability theorems for mapping class groups of surfaces were obtained in the eighties by Harer and Ivanov [14, 25] and recently turned out to be a key ingredient to a solution of the Mumford conjecture about the homology of the Riemann moduli space [30].”
The homotopy type of the cobordism category (2006, background): “This is the generalized Mumford conjecture, proved in [MW02].”
Homological stability for mapping class groups of surfaces (2010, methodology): “With our knowledge of the stable homology ([24, 26], see also [22]) and using recent calculations of Morita [27, Thm.”
Some problems on mapping class groups and moduli space (2006, background): “cular, for g≥7 determine whether or not Mod g is generated by 3 involutions. 5.2 Higher ﬁniteness properties and cohomology While there has been spectacular progress in understanding H∗(Mod g,Z) (see [MW]), very little is known about H∗(I g,Z), and even less about H∗(K g,Z). Further, we do not have ”
Homology of the moduli spaces and mapping class groups of framed, r‐Spin and Pin surfaces (2010, background): “Recent advances in the theory of moduli spaces of complex curves [14, 10] concern the stable topology of these spaces, that is, the topology of moduli spaces of curves of high genus.”