734 research outputs found
Secondary Characteristic Classes of Surface Bundles
The Miller-Morita-Mumford classes associate to an oriented surface bundle
a class . In this note we define for
each prime and each integer a secondary characteristic class
. The mod
reduction \lambda_i(E) \in H^*(B; \F_p) has zero indeterminacy and satisfies
.Comment: 6 page
The Equivariant Cobordism Category
For a finite group , we define an equivariant cobordism category
. Objects of the category are -dimensional closed
smooth -manifolds and morphisms are smooth -dimensional equivariant
cobordisms. We identify the homotopy type of its classifying space (i.e.
geometric realization of its simplicial nerve) as the fixed points of the
infinite loop space of an equivariant spectrum.Comment: 44 pages; v2 has more details and many improvement
Homological stability for moduli spaces of high dimensional manifolds. II
We prove a homological stability theorem for moduli spaces of manifolds of
dimension , for attaching handles of index at least , after these
manifolds have been stabilised by countably many copies of .
Combined with previous work of the authors, we obtain an analogue of the
Madsen--Weiss theorem for any simply-connected manifold of dimension .Comment: 60 pages, 4 figures. Final accepted versio
Tropical curves, graph complexes, and top weight cohomology of M_g
We study the topology of a space parametrizing stable tropical curves of
genus g with volume 1, showing that its reduced rational homology is
canonically identified with both the top weight cohomology of M_g and also with
the genus g part of the homology of Kontsevich's graph complex. Using a theorem
of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie
algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least
7. This disproves a recent conjecture of Church, Farb, and Putman as well as an
older, more general conjecture of Kontsevich. We also give an independent proof
of another theorem of Willwacher, that homology of the graph complex vanishes
in negative degrees.Comment: 31 pages. v2: streamlined exposition. Final version, to appear in J.
Amer. Math. So
Divisibility of the stable Miller-Morita-Mumford classes
We determine the sublattice generated by the Miller-Morita-Mumford classes
in the torsion free quotient of the integral cohomology ring of the
stable mapping class group. We further decide when the mod p reductions
vanish.Comment: 24 pages, 1 figur
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