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Archivio digitale delle tesi discusse presso l’Università di Pisa

Tesi etd-09022019-143142


Tipo di tesi
Tesi di laurea magistrale
Autore
CLINI, ANDREA
URN
etd-09022019-143142
Titolo
An Introduction to the Chern Conjecture
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Frigerio, Roberto
Parole chiave
  • affine manifold
  • characteristic class
  • Chern conjecture
  • Chern-Weil theory
  • Euler characteristic
Data inizio appello
20/09/2019
Consultabilità
Completa
Riassunto
The aim of the thesis is to give an introduction to the Chern conjecture: a long-standing conjecture which claims that closed affine manifolds have zero Euler characteristic.
We also give an exposition of the necessary background about characteristic classes, with a particular focus on Chern-Weil theory.
This is done by presenting some of the main (partial) results about the conjecture.
We first study affine manifolds and introduce the Euler class with its topological definition.
Later we focus on characteristic classes and classifying spaces.
We present some classical results by Benzécri, Milnor and Smillie, which solve the conjecture in dimension 2 and show that, in general dimensions, the hypotheses of the conjecture can not be weakened.
We finally focus on Chern-Weil theory and then present Kostant and Sullivan's proof of the conjecture in the case of complete affine manifolds.
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