Abstract
The classical Harer conjecture concerns the stable homology triviality of the evident embedding from braid groups to mapping class groups, a result established by Song and Tillmann. The main part of the proof is to show that the map induced on the plus construction of its classifying spaces (of their direct limits) is a double loop space map. In this talk, we present a proof of the generalized Harer conjecture, which extends the homology triviality to every (regular) embedding from braid groups to mapping class groups. The main strategy employed in the proof is to remove all the interchangeable subsurfaces and collapse the new boundary components. Then we get a covering space over a disk with marked points, which we know how to handle. The nal goal is to show that the map induced on their classifying spaces preserves the actions of the framed little 2-disks operad.
Content
Introduction
Schnelle Erklärung der Braid group und der Abbildungsklassengruppe und des Dehn-Twist.
Für eine Geschlossene Fläche sind die generatoren der Abbildungsklassengruppe die Humphrey Generatoren.
We do an embedding of a braid group into the Mapping class group.
The original Harer Conjecture states that the Homology of the infinite braid group and the infinite Mapping class group of the infinite genus surface is equal.
Segal und Tillman haben einen neuen Beweis vorgestellt.
Regular embedding
If is a reasonable surface and we have a automorphism the support are all points that are not fix. Beispiele:
- -fache Überlagerung
Wie können wir die Zöpfe als Abbildungsklassen des Torus interpretieren, indem wir die Löcher eines Genug -Torus Zopfartig bewegen.