84 research outputs found
Embedded Cobordism Categories and Spaces of Manifolds
Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the
classifying space of the cobordism category with objects (d-1)-dimensional
manifolds embedded in R^\infty. In this paper we apply the techniques of spaces
of manifolds, as developed by the author and Galatius, to identify the homotopy
type of the cobordism category with objects (d-1)-dimensional submanifolds of a
fixed background manifold M.
There is a description in terms of a space of sections of a bundle over M
associated to its tangent bundle. This can be interpreted as a form of Poincare
duality, relating a space of submanifolds of M to a space of functions on M
Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces
We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We
apply earlier work of the author to show that each of these moduli spaces
exhibits homological stability, and we identify the stable integral homology
with that of certain infinite loop spaces in each case. We further show that
these moduli spaces each have path components which are Eilenberg--MacLane
spaces for the framed, r-Spin and Pin mapping class groups respectively, and
hence we also identify the stable group homology of these groups.
In particular: the stable framed mapping class group has trivial rational
homology, and its abelianisation is Z/24; the rational homology of the stable
Pin mapping class groups coincides with that of the non-orientable mapping
class group, and their abelianisations are Z/2 for Pin^+ and (Z/2)^3 for Pin^-.Comment: 30 pages, 7 figures. V2: Revised to fit with arXiv:0909.4278. V3:
Submitted version. V4: Accepted version, to appear in Journal of Topolog
Generalised Miller-Morita-Mumford classes for block bundles and topological bundles
The most basic characteristic classes of smooth fibre bundles are the
generalised Miller-Morita-Mumford classes, obtained by fibre integrating
characteristic classes of the vertical tangent bundle. In this note we show
that they may be defined for more general families of manifolds than smooth
fibre bundles: smooth block bundles and topological fibre bundles.Comment: 18 page
The homology of the stable non-orientable mapping class group
Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and Korkmaz one
can identify the homotopy-type of the classifying space of the stable
non-orientable mapping class group (after plus-construction). At odd
primes p, the F_p-homology coincides with that of , but at
the prime 2 the result is less clear. We identify the F_2-homology as a Hopf
algebra in terms of the homology of well-known spaces. As an application we
tabulate the integral stable homology of in degrees up to six.
As in the oriented case, not all of these cohomology classes have a geometric
interpretation. We determine a polynomial subalgebra of
consisting of geometrically-defined characteristic classes.Comment: 15 pages; Section 6 completely revised, otherwise cosmetic change
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
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