The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces
نویسنده
چکیده
The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces 3 Introduction 3 Lecture 1. The Mumford conjecture and the Madsen-Weiss theorem 5 1. The Mumford conjecture 5 2. Moduli space, mapping class groups and diffeomorphism groups 5 3. The Mumford-Morita-Miller classes 7 4. Homological stability 7 5. The Madsen-Weiss theorem 9 6. Exercices 10 Lecture 2. Homological stability: geometric ingredients 11 1. General strategy of proof 11 2. The case of the mapping class group of surfaces 11 3. The ordered arc complex 12 4. Curve complexes and disc spaces 15 5. Exercices 15 Lecture 3. Homological stability: the spectral sequence argument 17 1. Double complexes associated to actions on simplicial complexes 17 2. The spectral sequence associated to the horizontal filtration 18 3. The spectral sequence associated to the vertical filtration 18 4. The proof of stability for surfaces with boundaries 19 5. Closing the boundaries 21 6. Exercises 21 Lecture 4. Homological stability: the connectivity argument 23 1. Strategy for computing the connectivity of the ordered arc complex 23 2. Contractibility of the full arc complex 24 3. Deducing connectivity of smaller complexes 25 4. Exercises 26 Bibliography 27 i The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces
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