Ulrike Tillmann

The stable mapping class group and Q(ℂP

Abstract.In [T2] it was shown that the classifying space of the stable mapping class groups after plus construction ℤ×BΓ+∞ has an infinite loop space structure. This result and the tools developed in [BM] to analyse transfer maps, are used here to show the following splitting theorem. Let Σ∞(ℂP∞+)∧p≃E0∨...∨Ep-2 be the “Adams-splitting” of the p-completed suspension spectrum of ℂP∞+. Then for some infinite loop space Wp,¶(ℤ×BΓ+∞)∧p≃Ω∞(E0)×...×Ω∞(Ep-3)×Wp¶where Ω∞Ei denotes the infinite loop space associated to the spectrum Ei. The homology of Ω∞Ei is known, and as a corollary one obtains large families of torsion classes in the homology of the stable mapping class group. This splitting also detects all the Miller-Morita-Mumford classes. Our results suggest a homotopy theoretic refinement of the Mumford conjecture. The above p-adic splitting uses a certain infinite loop map¶α∞:ℤ×BΓ+∞?Ω∞ℂP∞-1¶that induces an isomorphims in rational cohomology precisely if the Mumford conjecture is true. We suggest that α∞ might be a homotopy equivalence.

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.

Higher genus surface operad detects infinite loop spaces

Abstract. The operad studied in conformal field theory and introduced ten years ago by G. Segal [S] is built out of moduli spaces of Riemann surfaces. We show here that this operad which at first sight is a double loop space operad is indeed an infinite loop space operad. This leads to a new proof of the fact that the classifying space of the stable mapping class group

Stripping and splitting decorated mapping class groups

\mathbb Z \times B\Gamma_\infty ^+

Divisibility of the stable Miller-Morita-Mumford classes

, is an infinite loop space after plus construction [T2]. This new approach has various advantages. In particular, the infinite loop space structure is more explicid.

S-Structures for k-Linear Categories and the Definition of a Modular Functor

We study decorated mapping class groups, i.e., mapping class groups of surfaces with marked points and boundary components, and their behaviour under stabilization maps with respect to the genus, the number of punctures and boundary components. Decorated mapping class groups are non-trivial extensions of the undecorated mapping class group, and the first result states that the extension is homologically trivial when one stabilizes with respect to the genus. The second result implies that one also gets splittings of homology groups when stabilizing with respect to the number of punctures and boundary components.

Algebraic Topology: Applications and New Directions

We determine the sublattice generated by the Miller-Morita-Mumford classes

Embeddings of braid groups into mapping class groups and their homology

\kappa_i

On the Farrell cohomology of the mapping class group of non-orientable surfaces

in the torsion free quotient of the integral cohomology ring of the stable mapping class group. We further decide when the mod p reductions

Tubular configurations: equivariant scanning and splitting

\kappa_i