arXiv subjects

Ulrike Tillmann

Publications and source records attributed to Ulrike Tillmann.

30 records · Page 2Linked to original sources

arXiv2017

Infinite loop spaces from operads with homological stability

Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces. The recent, strong homological stability results of Galatius and Randal-Williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability. As a consequence the map to $K$-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.

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arXiv2017

A roadmap for the computation of persistent homology

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

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arXiv2016

Infinite loop spaces and nilpotent K-theory

Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces $BSU$, $BU$, $BSO$, $BO$, $BSp$, $BGL_{\infty}(R)^{+}$ and $Q_0(\mathbb{S}^{0})$. We show that these infinite loop spaces are the zero spaces of non-unital $E_\infty$-ring spectra. We introduce the notion of $q$-nilpotent K-theory of a CW-complex $X$ for any $q\ge 2$, which extends the notion of commutative K-theory defined by Adem-Gómez, and show that it is represented by $\mathbb Z\times B(q,U)$, were $B(q,U)$ is the $q$-th term of the aforementioned filtration of $BU$. For the proof we introduce an alternative way of associating an infinite loop space to a commutative $\mathbb{I}$-monoid and give criteria when it can be identified with the plus construction on the associated limit space. Furthermore, we introduce the notion of a commutative $\mathbb{I}$-rig and show that they give rise to non-unital $E_\infty$-ring spectra.

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arXiv2015

Homology stability for symmetric diffeomorphism and mapping class groups

For any smooth compact manifold $W$ of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of $k$ points or $k$ embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of $W$ connected sum with $k$ copies of an arbitrary compact smooth manifold $Q$ of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

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arXiv2013

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.

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arXiv2012

Embeddings of braid groups into mapping class groups and their homology

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists.

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arXiv2011

Vanishing of universal characteristic classes for handlebody groups and boundary bundles

Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more general theorem valid in all dimensions: universal characteristic classes made from monomials in the Pontrjagin classes (and even powers of the Euler class) vanish when pulled back from BDiff(\partial W) to BDiff(W).

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arXiv2010

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.

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arXiv2007

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.

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arXiv2007

Divisibility of the stable Miller-Morita-Mumford classes

We determine the sublattice generated by the Miller-Morita-Mumford classes $κ_i$ in the torsion free quotient of the integral cohomology ring of the stable mapping class group. We further decide when the mod p reductions $κ_i$ vanish.

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arXiv2003

Strings and the stable cohomology of mapping class groups

Twenty years ago, Mumford initiated the systematic study of the cohomology ring of moduli spaces of Riemann surfaces. Around the same time, Harer proved that the homology of the mapping class groups of oriented surfaces is independent of the genus in low degrees, increasing with the genus. The (co)homology of mapping class groups thus stabelizes. At least rationally, the mapping class groups have the same (co)homology as the corresponding moduli spaces. This prompted Mumford to conjecture that the stable rational cohomology of moduli spaces is generated by certain tautological classes that he defines. Much of the recent interest in this subject is motivated by mathematical physics and, in particular, by string theory. The study of the category of strings led to the discovery of an infinite loop space, the cohomology of which is the stable cohomology of the mapping class groups. We explain here a homotopy theoretic approach to Mumford's conjecture based on this fact. As byproducts infinite families of torsion classes in the stable cohomology are detected, and the divisibility of the tautological classes is determined. An analysis of the category of strings in a background space leads to the formulization of a parametrized version of Mumford's conjecture. The paper is chiefly a summary of the author's work and her collaboration with Ib Madsen. Earlier this year Madsen and Weiss announced a solution of Mumford's conjecture. We touch on some of the consequences and the ideas behind this most exciting new developement.

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arXiv1998

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.

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