80 research outputs found
S-Structures for k-linear categories and the definition of a modular functor
Motivated by ideas from string theory and quantum field theory new invariants
of knots and 3-dimensional manifolds have been constructed from complex
algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal
categories with additional structure (Turaev and Yetter), and modular functors
(Walker and Kontsevich). These constructions are very closely related. We take
a unifying categorical approach based on a natural 2-dimensional generalization
of a topological field theory in the sense of Atiyah and Segal, and show that
the axioms defining these complex algebraic structures are a consequence of the
underlying geometry of surfaces. In particular, we show that any linear
category over a field with an action of the surface category is semi-simple and
Artinian.Comment: Accepted for publication in the Journal of the LMS, April 199
Homological stability for oriented configuration spaces
We prove homological stability for sequences of "oriented configuration
spaces" as the number of points in the configuration goes to infinity. These
are spaces of configurations of n points in a connected manifold M of dimension
at least 2 which 'admits a boundary', with labels in a path-connected space X,
and with an orientation: an ordering of the points up to even permutations.
They are double covers of the corresponding unordered configuration spaces,
where the points do not have this orientation. To prove our result we adapt
methods from a paper of Randal-Williams, which proves homological stability in
the unordered case. Interestingly the oriented configuration spaces stabilise
more slowly than the unordered ones: the stability slope we obtain is
one-third, compared to one-half in the unordered case (these are the best
possible slopes in their respective cases). This result can also be interpreted
as homological stability for unordered configuration spaces with certain
twisted coefficients.Comment: 36 pages, 2 figures; v2: minor changes, final version - to appear in
Trans. Amer. Math. So
Tubular configurations: equivariant scanning and splitting
Replacing configurations of points by configurations of tubular
neighbourhoods (or discs) in a manifold, we are able to define a natural
scanning map that is equivariant under the action of the diffeomorphism group
of the manifold. We also construct the so-called power set map of configuration
spaces diffeomorphism equivariantly. Combining these two constructions yields
stable splittings in the sense of Snaith and generalisations thereof that are
equivariant. In particular one deduces stable splittings of homotopy orbit
spaces. As an application the homology injectivity is proved for diffeomorphism
of manifolds that fix an increasing number of points. Throughout we work with
configurations spaces with labels in a fibre bundle.Comment: 24 page
Braids, mapping class groups, and categorical delooping
Dehn twists around simple closed curves in oriented surfaces satisfy the
braid relations. This gives rise to a group theoretic from the braid group to
the mapping class group. We prove here that this map is trivial in stable
homology with any trivial coefficients. In particular this proves an old
conjecture of J. Harer. The main tool is categorical delooping. In an appendix
we discuss geometrically defined homomorphisms from the braid to the mapping
class group.Comment: 19 pages, 9 figures, late
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
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
The homotopy type of the cobordism category
The embedded cobordism category under study in this paper generalizes the
category of conformal surfaces, introduced by G. Segal in order to formalize
the concept of field theories. Our main result identifies the homotopy type of
the classifying space of the embedded d-dimensional cobordism category for all
d. For d=2, our results lead to a new proof of the generalized Mumford
conjecture, somewhat different in spirit from the original one.Comment: 40 pages. v2 has improved notation, added explanations, and minor
mistakes fixed. v3 has minor corrections and improvements. Final submitted
versio
Stratifying multiparameter persistent homology
A fundamental tool in topological data analysis is persistent homology, which
allows extraction of information from complex datasets in a robust way.
Persistent homology assigns a module over a principal ideal domain to a
one-parameter family of spaces obtained from the data. In applications data
often depend on several parameters, and in this case one is interested in
studying the persistent homology of a multiparameter family of spaces
associated to the data. While the theory of persistent homology for
one-parameter families is well-understood, the situation for multiparameter
families is more delicate. Following Carlsson and Zomorodian we recast the
problem in the setting of multigraded algebra, and we propose multigraded
Hilbert series, multigraded associated primes and local cohomology as
invariants for studying multiparameter persistent homology. Multigraded
associated primes provide a stratification of the region where a multigraded
module does not vanish, while multigraded Hilbert series and local cohomology
give a measure of the size of components of the module supported on different
strata. These invariants generalize in a suitable sense the invariant for the
one-parameter case.Comment: Minor improvements throughout. In particular: we extended the
introduction, added Table 1, which gives a dictionary between terms used in
PH and commutative algebra; we streamlined Section 3; we added Proposition
4.49 about the information captured by the cp-rank; we moved the code from
the appendix to github. Final version, to appear in SIAG
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