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In some sense, this chapter complements a chapter by G. Mondello in Volume II of this Handbook which gives a detailed account of the use of ribbon graphs in the intersection theory of moduli space, in relation to the Witten conjecture. 2 The generalized Mumford conjecture In Chapter 12, Ib Madsen gives a survey of the proof of a generalized version of the Mumford conjecture which he obtained in joint work with M. Weiss. The original Mumford conjecture states that the stable rational cohomology of the moduli space Mg is a certain polynomial algebra generated by the Mumford–Morita–Miller cohomology classes of even degrees.
37 41 41 42 43 44 47 53 54 56 56 57 61 63 65 65 67 68 1 Introduction The analytic theory of Teichmüller space is based on the notion of quasiconformal homeomorphisms that is inherent in the definition. Let us recall this definition. e. x; y/ j y > 0g by Möbius transformations. A quasiconformal deformation of is a pair . ; /, where is a representation of into PSL(2; R/ and W H !
Z; ı; r/, is bigger than K 2 . z; ı; r/ is bigger than 2 . z 0 /. zi ; zj / and . z 0 //. 1) (cf. A0 / Ä Á2 ; therefore Á Ä , so that d. z; ı/ and for all z 2 K. Therefore f n g is equicontinuous. 6. 1 one could prove that the n are Hölder continuous on K, with an exponent depending only on K. By a theorem of Akira Mori this exponent can be taken equal to K1 . Up to extracting a subsequence, we suppose now that . n / converges to uniformly on compact sets in U . (ii) Any point z 2 U has a neighborhood on which is either injective or constant.
Datacomm Basics by Veronica Williams