TY - JOUR
T1 - Homological stability and stable moduli of flat manifold bundles
AU - Nariman, Sam
N1 - Funding Information:
I would like to thank my advisor, Søren Galatius, for his guidance. Without his help and his encouragement, this article would have never existed. I would also like to thank Oscar Randal-Williams for many helpful discussions, in particular I learned the method of “relative” spectral sequence from him. I also owe Cary Malkiewich for his helpful comments on the proof of Theorem 5.13 . I would like to thank Steve Hurder for pointing out to me that there are more continuously varying secondary classes than I wrote in the first draft. I would also like to thank Alexander Kupers and the referee for their careful reading and detailed comments on the first draft that made me considerably improve the paper. This work was supported in part by NSF grants DMS-1105058 and DMS-1405001 .
Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/11/7
Y1 - 2017/11/7
N2 - We prove that the group homology of the diffeomorphism group of #gSn×Sn\int(D2n) as a discrete group is independent of g in a range, provided that n>2. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford–Morita–Miller classes lift to a secondary R/Z-invariants for flat (#gSn×Sn)-bundles provided g≫0.
AB - We prove that the group homology of the diffeomorphism group of #gSn×Sn\int(D2n) as a discrete group is independent of g in a range, provided that n>2. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford–Morita–Miller classes lift to a secondary R/Z-invariants for flat (#gSn×Sn)-bundles provided g≫0.
KW - Diffeomorphism groups
KW - Flat bundles
KW - Godbillon–Vey invariants
KW - Haefliger classifying space
KW - Homological stability
KW - Infinite loop space
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U2 - 10.1016/j.aim.2017.09.015
DO - 10.1016/j.aim.2017.09.015
M3 - Article
AN - SCOPUS:85032199181
SN - 0001-8708
VL - 320
SP - 1227
EP - 1268
JO - Advances in Mathematics
JF - Advances in Mathematics
ER -