2,698 research outputs found
Generating the Johnson filtration
For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of
the mapping class group of a genus g surface with one boundary component. We
prove that for all k, there exists some G_k >= 0 such that Torelli_g^1(k) is
generated by elements which are supported on subsurfaces whose genus is at most
G_k. We also prove similar theorems for the Johnson filtration of Aut(F_n) and
for certain mod-p analogues of the Johnson filtrations of both the mapping
class group and of Aut(F_n). The main tools used in the proofs are the related
theories of FI-modules (due to the first author together with Ellenberg and
Farb) and central stability (due to the second author), both of which concern
the representation theory of the symmetric groups over Z.Comment: 32 pages; v2: paper reorganized. Final version, to appear in Geometry
and Topolog
On finite generation of the Johnson filtrations
We prove that every term of the lower central series and Johnson filtrations
of the Torelli subgroups of the mapping class group and the automorphism group
of a free group is finitely generated in a linear stable range. This was
originally proved for the second terms by Ershov and He.Comment: 32 pages. v2: very minor edits. Weaker versions of the results of
this paper previously appeared in arXiv:1704.01529 and arXiv:1703.04190v
Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)
We prove a new structural result for the spherical Tits building attached to
SL_n(K) for many number fields K, and more generally for the fraction fields of
many Dedekind domains O: the Steinberg module St_n(K) is generated by integral
apartments if and only if the ideal class group cl(O) is trivial. We deduce
this integrality by proving that the complex of partial bases of O^n is
Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results
for H^{vcd}(SL_n(O_K); Q), where O_K is the ring of integers in a number field
and vcd is the virtual cohomological dimension of SL_n(O_K). The (non)vanishing
depends on the (non)triviality of the class group of O_K. We also obtain a
vanishing theorem for the cohomology H^{vcd}(SL_n(O_K); V) with twisted
coefficients V.Comment: 36 pages; final version; to appear in Amer. J. Mat
A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)
In this paper we conjecture the stability and vanishing of a large piece of
the unstable rational cohomology of SL_n Z, of mapping class groups, and of
Aut(F_n).Comment: 18 pages. v2: final version, to appear in Algebraic Topology:
Applications and New Directions, AMS Contemporary Mathematics Serie
On the geometric nature of characteristic classes of surface bundles
Each Morita--Mumford--Miller (MMM) class e_n assigns to each genus g >= 2
surface bundle S_g -> E^{2n+2} -> M^{2n} an integer e_n^#(E -> M) :=
in Z. We prove that when n is odd the number e_n^#(E -> M) depends only on the
diffeomorphism type of E, not on g, M, or the map E -> M. More generally, we
prove that e_n^#(E -> M) depends only on the cobordism class of E. Recent work
of Hatcher implies that this stronger statement is false when n is even. If E
-> M is a holomorphic fibering of complex manifolds, we show that for every n
the number e_n^#(E -> M) only depends on the complex cobordism type of E.
We give a general procedure to construct manifolds fibering as surface
bundles in multiple ways, providing infinitely many examples to which our
theorems apply. As an application of our results we give a new proof of the
rational case of a recent theorem of Giansiracusa--Tillmann that the odd MMM
classes e_{2i-1} vanish for any surface bundle which bounds a handlebody
bundle. We show how the MMM classes can be seen as obstructions to low-genus
fiberings. Finally, we discuss a number of open questions that arise from this
work.Comment: 26 pages. v2: added examples to final section; v3: improved main
theorem for complex fiberings; v4: final version, to appear in Journal of
Topolog
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