the webpage of Danica Kosanović

# Danica Kosanović

Pronounce c as zz in pizza, and ć as ci in ciabatta. Typeset the letter ć as \'c in LaTeX.
On some devices you are offered ć as an option after long pressing c.

ETH Zürich, Forschungsinstitut für Mathematik
Rämistrasse 101, HG GO 68.2
8092 Zürich, Switzerland

danica.kosanovic[at]math[dot]ethz[dot]ch

I am a Hermann Weyl Instructor at the Departement of Mathematics at ETH Zürich (Swiss Federal Institute of Technology in Zurich). My mentors are Peter Feller and Thomas Willwacher

My interests include knot theory, 4-manifolds, knotted surfaces in 4-manifolds, homotopy types of embedding spaces, Goodwillie-Weiss embedding calculus, operads, graph complexes. For more details, see the slides from the public talk of my PhD defense, or have a look at the tabs on the left.

Before coming to ETH in 2021, I was a FSMP postdoc at Paris 13 (Université Sorbonne Paris Nord), working with Geoffroy Horel. I obtained my PhD degree in September 2020 from the University of Bonn (Germany), working at the Max-Planck Insitut für Mathematik under the supervision of Peter Teichner. Previously, I studied in Belgrade (Serbia) and Cambridge (UK).

My partner Mihajlo Cekić is also a mathematician.

### News

 September 2022 The preprint "A space level light bulb theorem for disks" with Peter Teichner has now been split into two papers, both considerably improved and expanded, see arxiv.org/abs/2105.13032 (space level, all dimensions) and arxiv.org/abs/2209.12015 (results in dimension 4).

### Upcoming

 Autumn 2022 I am teaching Topology of Manifolds in the autumn semester (HS22) @ ETH, see the tab Teaching. 17.11.2022 2-knots and knotted families of arcs, University of Zurich, room Y27-H-26, at 10:30

I like thinking about knots, 4-manifolds, surfaces inside, and in general about topology in low dimensions! However, I also believe that formalism and tools of higher topology, i.e. homotopy theory, higher categories, TQFT’s, operads, as well as combinatorics of Feynman diagrams and configuration spaces, can merge together to give even more insight about low-dimensional manifolds.

In my thesis I studied finite type knot invariants and their relation to the Goodwillie-Weiss embedding calculus.

### Preprints

 Embedding calculus and grope cobordism of knots arxiv.org/abs/2010.05120 (submitted) We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary 3-manifold. This solves some remaining open cases of Goodwillie–Klein–Weiss connectivity estimates, and at the same time confirms one half of the conjecture that for classical knots $ev_n$ are universal additive Vassiliev invariants over the integers. In addition, we give a sufficient condition for this conjecture to hold over a coefficient group, which is by recent results of Boavida de Brito and Horel fulfilled for the rationals and for the $p$-adic integers in a range. Therefore, embedding calculus invariants are strictly more powerful than the Kontsevich integral. Furthermore, our work shows they are more computable as well. Namely, the main theorem computes the first possibly non-vanishing invariant $ev_n$ of a knot which is grope cobordant to the unknot to be precisely equal to the equivalence class of the underlying decorated tree of the grope in the associated graph complex. Actually, our techniques apply beyond dimension $3$, offering a description of the layers in embedding calculus for long knots in a manifold of any dimension, and suggesting that certain generalised gropes realise the corresponding graph complex classes. A space level light bulb theorem in all dimensions ~ Joint with Peter Teichner ~ arxiv.org/abs/2105.13032 (submitted) Given a $d$-dimensional manifold $M$ and a knotted sphere $s\colon\mathbb{S}^{k-1}\hookrightarrow\partial M$ with $1\leq k\leq d$, for which there exists a framed dual sphere $G\colon\mathbb{S}^{d-k}\hookrightarrow\partial M$, we show that the space of neat embeddings $\mathbb{D}^k\hookrightarrow M$ with boundary $s$ can be delooped by the space of neatly embedded $(k-1)$-disks, with a normal vector field, in the $d$-manifold obtained from $M$ by attaching a handle to $G$. This increase in codimension significantly simplifies the homotopy type of such embedding spaces, and is of interest also in low-dimensional topology. In particular, we apply the work of Dax to describe the first interesting homotopy group of these embedding spaces, in degree $d-2k$. In a separate paper we use this to give a complete isotopy classification of 2-disks in a 4-manifold with such a boundary dual. A new approach to light bulb tricks: Disks in 4-manifolds ~ Joint with Peter Teichner ~ arxiv.org/abs/2209.12015 (submitted) For a 4-manifold $M$ and a knot $k\colon\mathbb{S}^1\hookrightarrow\partial M$ with dual sphere $G\colon\mathbb{S}^2\hookrightarrow\partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings $\mathbb{D}^2\hookrightarrow M$ with boundary $k$, using an invariant going back to Dax. Moreover, we construct a group structure on $\mathbb{D}(M;k)$ and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group $\mathbb{D}(M;k)$ to the mapping class group of $M$. On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant arxiv.org/abs/2111.03041 accepted for publication in the Transactions of the American Mathematical Society accepted version (pdf) We compute in many classes of examples the first potentially interesting homotopy group of the space of embeddings of either an arc or a circle into a manifold M of dimension $d\geq 4$. In particular, if M is a simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group of M, answering a question posed by Arone and Szymik. The case $d = 3$ gives isotopy invariants of knots in a 3-manifold, that are universal of Vassiliev type $\leq 1$, and reduce to Schneiderman’s concordance invariant.

### Thesis

 A geometric approach to the embedding calculus knot invariants. PhD Thesis. Download in Bonn Library. Finite type invariants (often called Gusarov-Vassiliev, or just Vassiliev, invariants) give a certain filtration on the set of all invariants by their type. A dual point of view is, however, more geometric: there is a filtration on the monoid of knots itself, which arises by looking at a certain sequence of n-equivalence relations on knots. Then the n-th term of the filtration is comprised of knots which are n-equivalent to the unknot. For example, two knots are 1-equivalent if they can be related by a sequence of crossing changes. This means that the first term in the filtration is equal to the whole monoid of knots! To get an idea about 2-equivalence, take a look at the operation on the left - grab some three strands of a knot and connect-sum them with the Borromean rings. Embedding calculus of Goodwillie and Weiss is another homotopy-theoretic approach to spaces of embeddings. When applied to the embedding functor of long knots $\mathcal{K}$ in the 3-space it yields a tower of spaces $T_n$ together with evaluation maps $ev_n\colon K\to T_n$. These spaces turn out to be very interesting. For example, they can be shown to be double loop spaces of the mapping spaces between some (truncated) operads. Hence, their components form an abelian group and the evaluation map from knots gives a map on $\pi_0$ which turns out to be a finite type invariant! It is conjectured to be universal such, in other words, the group of knots modulo relation of n-equivalence is isomorphic to $\pi_0T_n$. Therefore, the two stories should not be so separate after all. One unifying perspective is that of gropes. Namely, the trivalent vertices appearing in the diagrams for finite type theory (originating in quantum Chern-Simons theory) correspond to the Borromean rings, and the isotopy depicted below hints at how this in turn relates to gropes. In the very last picture we clearly see a genus one surface with one boundary component emerging. This will represent the bottom stage of a grope.

### Research talks

 04.11.2022 A new approach to light bulb tricks @ BIRS Workshop Topology in Dimension 4.5, Banff 24.-27.10.2022 tba @ Réunion annuelle du GDR, 24-27 October 2022, Université de Nantes, webpage, slides 14.09.2022 2-knots and knotted families of arcs @ DMV (German Mathematical Society) Annual Meeting, Berlin, beamer slides 16.07.2022 2-knots and knotted families of arcs @ AMS-EMS-SMF Congress of Mathematics, 18-22 July, 2022, Grenoble 25.03.2022 A light bulb theorem for disks @ Algebraic Geometry and Differential Topology Seminar, Alfréd Rényi Institute of Mathematics, Budapest 24.02.2022 Smooth embeddings and their families @ Durham Geometry and Topology Seminar 16.02.2022 Smooth embeddings and their families @ Cambridge Differential Geometry and Topology Seminar 11.02.2022 Smooth embeddings and their families @ UL Lafayette Topology Seminar 7.10.2021 - Online A light bulb theorem for disks @ Princeton Topology Seminar 6.10.2021 Light bulbs in 4-manifolds @ ETH Zürich Geometrie Seminar 21.9.2021 - Online A light bulb theorem for disks @ University of Virginia Geometry Seminar 17.8.2021 Учворени дискови у четири димензије @ Workshop on Symplectic Topology, University of Belgrade 9.7.2021 - Online Higher homotopy groups in low dimensional topology @ Young Topologists Meeting, beamer slides 11.6.2021 - Online A light bulb theorem for disks @ Georgia Topology Conference, beamer slides 21.4.2021 - Online Knotted families of arcs @ Münster Topology Seminar 15.3.2021 - Online Knotted families of arcs @ MIT Topology Seminar 13.1.2021 - Online Knot invariants from homotopy theory @ Higher Structures & Field Theory Seminar 4.12.2020 - Online Knot invariants from homotopy theory @ Colloquium LAGA Paris 13 3.12.2020 - Online Knot invariants from homotopy theory @ Théorie des groupes, LAMFA Université d'Amiens 26.11.2020 - Online Knot invariants from homotopy theory @ Séminaire AGATA, Université de Montpellier, beamer slides 17.11.2020 - Online Knot invariants from homotopy theory @ Warwick algebraic topology seminar 2.11.2020 - Online Knot invariants from homotopy theory @ G&T Seminar Glasgow 16.10.2020 Knot invariants from homotopy theory @ Université de Lille 31.7.2020 - Online Embedding calculus for knot spaces @ Oberwolfach Workshop Topologie 29.5.2020 - Online Knot invariants from homotopy theory @ Topological Quantum Field Theory Seminar, Instituto Superior Técnico, Lisboa, video 21.4.2020 - Online Knot invariants from homotopy theory @ jointly Séminaire de l'équipe Topologie Algébrique, LAGA, Paris 13 and Séminaire de Topologie, IMJ-PRG, Paris 7 20.2.2020 A geometric approach to the embedding calculus @ Oberwolfach Workshop Low-dimensional Topology 30.1.2020 Knot invariants from homotopy theory @ Topology Seminar Bochum 20.1.2020 Knot theory meets the embedding calculus @ Copenhagen Algebra/Topology Seminar 16.1.2020 Нове технике у теорији утапања (New techniques in the theory of embeddings) @ Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade 2.12.2019 Knot theory meets the embedding calculus @ MPIM Topology Seminar, Bonn 16.10.2019 Knots map onto components of the embedding calculus tower @ Spaces of Embeddings: Connections and Applications, Banff International Research Station, Canada 16.9.2019 A gong show talk @ Workshop on 4-manifolds, MPIM Bonn 13.5.2019 A gong show talk @ Knots and Braids in Norway (KaBiN), Trondheim 7.5.2019 A geometric approach to embedding calculus @ Utrecht Geometry Center Seminar 25.12.2018 Инваријанте чворова и конфигурациони простори (Knot invariants and configuration spaces) @ Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, slides (in Serbian) 17.12.2018 Revisiting the Arf invariant @ Topology Seminar, MPIM Bonn 6.12.2018 Extended evaluation maps from knots to the embedding tower @ Manifolds Workshop (part of Homotopy Harnessing Higher Structures Trimester) at Isaac Newton Institute, Cambridge 28.11.2018 Knot theory meets homotopy theory @ IMPRS Seminar, MPIM Bonn, slides 24.7.2018 Grope cobordism and the embedding tower for knots @ ICM 2018 Satellite Conference: Braid Groups, Configuration Spaces and Homotopy Theory, in Salvador, Brazil Feb 2018 - Poster A homotopy theoretic approach to finite type knot invariants @ Winter Braids, CIRM, Luminy, France

### Expository talks

 25.5.2022 Freedman's work on 4-manifolds @ Actualité des mathématiques, Neuchâtel 6.5.2021 On a theorem of Kontsevich and Conant-Vogtmann @ Groupes de Grothendieck-Teichmüller et applications 11.3.2021 Chord diagram invariants of tangles @ Groupes de Grothendieck-Teichmüller et applications, notes 13.2.2020 - Online On the punctured knots model for embedding spaces @ Configuration Categories Learning Seminar 19.12.2019 - Online On link maps @ Mojito’s Seminar 13.2.2020 - Online On the paper by Bundey-Gabai about knotted 3-balls @ Online Student Seminar, notes 19.12.2019 Watanabe's counting formula for classes in Diff(S^4) @ Hot Topic Seminar, MPIM 5.11.2019 Milnor invariants and Whitney towers @ Milnor Invariants Learning Seminar, MPIM July 2019 Introduction to Milnor link invariants and relation to Massey products @ Milnor Invariants Learning Seminar, MPIM May 2019 Formality of little disks operads @ IMPRS seminar, MPIM Sep/Oct 2018 Two talks about the paper of Ihara on automorphisms of pure sphere braid group @ GT learning seminar, MPIM Apr/May 2018 Two talks on perturbative quantization and Chern-Simons theory for knots @ BV learning seminar, MPIM 22.3.2018 Complex oriented cohomology theories @ Peter’s Seminar in Berkeley 06.12.2017 Universal Knot Invariants @ The Chinese University of Hong Kong 15.11.2017 How to draw a smooth 4−manifold? @ IMPRS seminar, MPIM 25.09.2017 A categorical approach to quantum knot invariants @ Topology Seminar, MPIM 04.08.2017 A survey of Witten-Reshetikhin-Turaev invariants of 3-manifolds @ Special Topology Seminar, MPIM 02.06.2017 Topological reincarnations of the Arf invariant @ Cambridge Junior Geometry Tea Seminar, Cambridge, UK 23.03.2017 Topological reincarnations of the Arf invariant @ Berkeley seminar

## Topology of Manifolds

(Autumn 2022)

#### Announcements.

• The official syllabus.
• The weekly schedule (pdf, updated 22.11.2022).
• Lecture notes are uploaded after each class, see below. (Note the large file size!)
• Exercise classes are held in Weeks 2, 4, 5, 9, 11, 13 (Monday 16:15 at HG E 33.5).
• Exercises sheets are published on this webpage on Tuesdays two weeks before (see below).
• Solutions of exercises are published after the respective exercise class. They are prepared by the course coordinator: Miguel Moreira, miguel.moreira[at]math[dot]ethz[dot]ch.
• The exam will be written during the last lecture slot.
• place: ML F 39
• date: 20.12.2022
• time: starting at 16:30 ending at 18:00 (90 minutes).
• PhD students who plan to regularly attend the course should register for it in eDoz in order to earn credits (choose the -DRL course).
• There will be NO LECTURE on October 25. Instead, in the week after there will be two lectures (in the slot, and room, of the exercise class on October 31, and in the regular slot on November 1).