# Florian Kranhold

Address Mathematical Institute University of Bonn Endenicher Allee 60 53115 Bonn 4.020 +49 (0)228 73-62208 PGP Public Key (0xD2890F65)

I am a third year PhD student of Carl-Friedrich Bödigheimer. My PhD project deals with various coloured topological operads, which allow arguments of higher multiplicity. Particular examples are clustered versions of May’s little cubes or of Tillmann’s surface operad. On the one hand, their free algebras are certain clustered configuration spaces which have been studied recently. On the other hand, they act on moduli spaces of Riemann surfaces with multiple boundary curves. This gives rise to a collection of operations on the homology of moduli spaces which help us to understand unstable classes, and may also exhibit variations of the classical moduli spaces as an infinite loop space in the spirit of Madsen and Weiss.

### Teaching

In the upcoming summer term 2021, I will be tutor for the lecture Algebraic Topology 2 given by Carl-Friedrich Bödigheimer. In the summer term 2020, I co-organised a graduate seminar on Operads in Algebra and Topology together with Andrea Bianchi. Moreover, I have been tutor for the following topology lectures:

 WiSe 2020/21 Algebraic Topology 1 Carl-Friedrich Bödigheimer SoSe 2020 Algebraic Topology 2 Christoph Winges WiSe 2019/20 Algebraic Topology 1 Wolfgang Lück SoSe 2019 Topology 2 Daniel Kasprowski WiSe 2018/19 Topology 1 Wolfgang Lück SoSe 2018 Einführung in die Geometrie und Topologie Wolfgang Lück

### Preprints

• #### Parametrised moduli spaces of surfaces as infinite loop spaces

We study the $$E_2$$-algebra $$\Lambda\mathfrak{M}_{*,1}=\coprod_{g\geqslant 0}\Lambda\mathfrak{M}_{g,1}$$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $$\Omega B\Lambda\mathfrak{M}_{*,1}$$: it is the product of $$\Omega^\infty\mathbf{MTSO}(2)$$ with a certain free $$\Omega^\infty$$-space depending on the family of all boundary-irreducible mapping classes in all mapping class groups $$\Gamma_{g,n}$$ with $$g\geqslant 0$$ and $$n\geqslant 1$$.

• #### Configuration spaces of clusters as $$E_d$$-algebras

It is a classical result that configuration spaces of labelled particles in $$\mathbb{R}^d$$ are free algebras over the little $$d$$-cubes operad $$\mathscr{C}_d$$, and their $$d$$-fold bar construction is equivalent to the $$d$$-fold suspension of the labelling space. The aim of this paper is to study a variation of these spaces, namely the configuration space of labelled clusters of points in $$\mathbb{R}^d$$. This configuration space is again an $$E_d$$-algebra, but in general not a free one. We give geometric models for their iterated bar construction in two different ways: one uses an additional verticality constraint, and the other one uses a description of these clustered configuration spaces as cellular $$E_1$$-algebras. In the last section, we show a stable splitting result and present some applications.

• #### Vertical configuration spaces and their homology

We introduce ordered and unordered configuration spaces of ‘clusters’ of points in an Euclidean space $$\mathbb{R}^d$$, where points in each cluster have to satisfy a ‘verticality’ condition, depending on a decomposition $$d=p+q.$$ We compute the homology in the ordered case and prove homological stability in the unordered case.

• #### Moduli spaces of Riemann surfaces and symmetric products: A combinatorial description of the Mumford–Miller–Morita classes

This is my master thesis which was finished in the summer term 2018 under the supervision of Carl-Friedrich Bödigheimer. Its main result is a Poincaré–Lefschetz correspondene between the Mumford–Miller–Morita classes in the cohomology of moduli spaces and certain simplicial subcomplexes of Bödigheimer’s model of parallel slit domains.