Winter Term 2019/2020

Lecturer: Dr. Gabriel Angelini-Knoll, Prof. Dr. Holger Reich, Prof. Dr. Elmar Vogt

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• Time and place:  Tuesday 14:00-16:00 h, Arnimallee 6, seminar room 009

• Leistungsnachweis/ criteria for proof of performance:  Presentation and written summary.

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Target group: The seminar is intended for students who have taken a first course in Topology.

Requirements: Basic terms from Topology I. Knowledge of basic terms from Topology II will be helpful, but not necessary. Participants of the lecture Topologie II, which takes place this semester, are especially welcome.

Content:  Simplicial methods have been intensively used in algebraic topology since the mid 1950s, but also play an ever increasing role in other fields of mathematics, unified in the concept of higher category theory. The slogan for topologists is that one can do more or less all of algebraic topology without ever looking at a topological space. Instead of topological spaces one looks at the category of simplicial sets. A simplicial set is a purely combinatorial object consisting for each non negative integer $n$ of a set, called the set of $n$-simplices, together with maps associating with any $n$-simplex and $0 \leq i \leq n$ its $i$-th face, an $(n-1)$-simplex, and its $i$-th degeneracy, an $(n+1)$-simplex.These maps have to satisfy certain obvious commutation relations. Simplicial maps between simplicial sets are then defined in the obvious way.
Using this simple structure one can define homotopies between such maps and homotopy and homology groups of simplicial sets, at least if the simplicial set satisfies a certain extension condition. An important example is the simplicial set associated to any topological space whose $n$-simplices are the continuous maps from the standard $n$-simplex into the space. This relates the category of topological spaces with the category of simplicial sets.
The purpose of the seminar is threefold. One is to familiarize the participants with working with simplicial sets. This aspect is only briefly touched, if at all, in the usual topology lecture cycle. The second is to develop homotopy concepts purely in simplicial sets, and the third is to relate all this to the usual homotopy and homology theory of topological spaces.
The talks should be accessible to any advanced undergraduate, but so that talks can be prepared early on, the later talks should be taken by participants who are already familiar with some basic homological concepts like chain complexes and chain homotopy.
To get a first impression, please read paragraphs 1-2 and 3.1 and 3.2 of paragraph 3 of the reference below. This will facilitate our discussion when we meet on October 15 for the first time. At this date we will also give a short overview of the planned talks and assign speakers to them.

The seminar language is English.

Schedule

Termin	Titel	Vortragende(r)
15.10.	Vorbesprechung / preliminary discussion	Gabriel Angelini-Knoll Holger Reich
22.10.	Homotopy Groups of Simplicial Sets: §3 and §4 of book	Robin Chemnitz
29.10.	Homotopy Groups of Simplicial Sets	Robin Chemnitz
	Homotopy of Simplicial Maps and Function Complexes I	Elisabeth a.d. Siepen
05.11.	Homotopy of Simplicial Maps and Function Complexes I: §5 and 6.1 to 6.4 of book	Elisabeth a.d. Siepen
12.11.	Homotopy of Simplicial Maps and Function Complexes II: rest of §6 of book	Dennis Chemnitz
19.11.	Homotopy of Simplicial Maps and Function Complexes II	Dennis Chemnitz
	Kan Fibrations	Vira Raichenko
26.11.	Kan Fibrations: §7 of book	Vira Raichenko
03.12.	Postnikov Systems and Minimal Sets : §8 and §9 of book	Kemal Rose
10.12.	Minimal Fibrations: §10 of book	Fabian Gringel
17.12.	Weak homotopy type and the Hurewicz Theorems: §12 and §13 of book	Evgeniya Lagoda
2020
07.01.	Geometric Realization of Simplicial Sets: §14 of book	Ferry Saavedra
14.01.	Relating the categories of Simplicial Sets and of Topological Spaces: §15 (recall of adloint functors) and §16	Ferry Saavedra
21.01.
28.01.
04.02.
11.02.

Literature:

• J. Peter May: Simplicial Objects in Algebraic Topology, University of Chicago Press,1982.
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