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Fakultät für Mathematik Universität Regensburg
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Simple homotopy theory and Whitehead groups
SoSe 2019

Georgios Raptis

Type of course (Veranstaltungsart)

Algebraic topology is generally concerned with the notion of homotopy equivalence. This Seminar will be about a combinatorial approach to the definition of homotopy equivalence for finite CW complexes. We will ask the following fundamental question: is there is a collection of elementary geometrically defined homotopy equivalences from which every homotopy equivalence is generated?

The definition of simple homotopy equivalence is based on such elementary homotopy equivalences, which act as building blocks or series of moves. The question then becomes: is every homotopy equivalence between finite CW complexes simple?

The answer turns out to be "no", but in a very interesting way. There is a single algebraic invariant, called the Whitehead torsion, which decides whether a homotopy equivalence is simple. The Whitehead torsion of a homotopy equivalence is an element of an abelian group, called the Whitehead group, which depends only on the fundamental group.

In this Seminar, we will first introduce and study simple homotopy equivalences, then we will define Whitehead groups and discuss their properties, and finally, we will prove the relationship between the two and answer the questions stated above. This answer is a beautiful (and rare) instance where topology and algebra match up exactly.

The Seminar should be of interest to those interested in algebraic topology. The topic can also serve as an introduction to algebraic K-theory from a topological viewpoint.

Depending on the interest in the topic, there may be a continuation of the Seminar next semester about geometric applications of simple homotopy theory and connections with differential topology (e.g. the s-cobordism theorem).

The seminar talks will be in English or German.

The main reference for this Seminar is the book:
M. M. Cohen "A Course in Simple-Homotopy Theory"

Recommended previous knowledge
Basic algebraic topology (covering spaces, CW complexes, singular and cellular homology) and some general familiarity with commutative algebra.

Dienstag 10-12


Course homepage
Details about the Seminar (e.g. schedule, notes, etc.) will appear on GRIPS and here http://graptismath.net/teaching.html
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

  • Organisational meeting/distribution of topics: Die Vorbesprechung für das Seminar findet
    am Freitag 01.03.2019, 10-12, M 228, statt. Wenn Sie nicht zur Vorbesprechung kommen
    können, melden Sie sich bitte per Email.
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Presentation: Giving a seminar talk of roughly 90 minutes
Examination (Prüfungsleistungen)
  • Detailed written report of the seminar talk
Regelungen bei Studienbeginn vor WS 2015 / 16
  • Benotet:
    • O. g. Studienleistung und o. g. Prüfungsleistung; die Note ergibt sich aus dem Seminarvortrag
  • Unbenotet:
    • O. g. Studienleistung
Additional comments
Mit Repetitorium, jeweils in der Woche vor den Vorträgen. Termin verhandelbar.

BV, BSem, MV, MSem, LA-GySem

Siehe Modulkatalog. MV und Nebenfach: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn
vor WS 15/16