TAILIEUCHUNG - Đề tài " The stable moduli space of Riemann surfaces: Mumford’s conjecture "

D. Mumford conjectured in [33] that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes κi of dimension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by BΓ∞ , where Γ∞ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of “large” genus. Tillmann’s theorem [44] that the plus construction makes BΓ∞ into an infinite loop space led to a stable homotopy version of Mumford’s conjecture, stronger than the original [24]. . | Annals of Mathematics The stable moduli space of Riemann surfaces Mumford s conjecture By Ib Madsen and Michael Weiss Annals of Mathematics 165 2007 843 941 The stable moduli space of Riemann surfaces Mumford s conjecture By IB Madsen and Michael Weiss Abstract D. Mumford conjectured in 33 that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes Ki of dimension 2i. For the purpose of calculating rational cohomology one may replace the stable moduli space of Riemann surfaces by Bl where r is the group of isotopy classes of automorphisms of a smooth oriented connected surface of large genus. Tillmann s theorem 44 that the plus construction makes Bl into an infinite loop space led to a stable homotopy version of Mumford s conjecture stronger than the original 24 . We prove the stronger version relying on Harer s stability theorem 17 Vassiliev s theorem concerning spaces of functions with moderate singularities 46 45 and methods from homotopy theory. Contents 1. Introduction Results and methods . Main result . A geometric formulation . Outline of proof 2. Families sheaves and their representing spaces . Language . Families with analytic data . Families with formal-analytic data . Concordance theory of sheaves . Some useful concordances 3. The lower row of diagram . A cofiber sequence of Thom spectra . The spaces hw and hV . The space hwloc . The space wloc . partially supported by American Institute of Mathematics. . partially supported by the Royal Society and by the Engineering and Physical Sciences Research Council Grant GR R17010 01. 844 IB MADSEN AND MICHAEL WEISS 4. Application of Vassiliev s -principle . Sheaves with category structure . Armlets . Proof of Theorem 5. Some homotopy colimit decompositions . Description of main results . Morse singularities Hessians and surgeries . Right-hand column . Upper left-hand column

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