Arbeitsgemeinschaft mit aktuellem Thema: Algebraic cobordism (Q2577095)

Summary: The aim of this Arbeitsgemeinschaft was to present the theory of algebraic cobordism due to Marc Levine and Fabien Morel through the lines of their original articles: Inspired by the work of Quillen on complex cobordism, one first introduces the notion of oriented cohomology theory on the category of smooth varieties over a field \(k\). Grothendieck's method allows one to extend the theory of Chern classes to such theories. When char\((k) = 0\), one proves the existence of a universal oriented cohomology theory \(X\to \Omega^*(X)\). Localisation and homotopy invariance are then proved for this universal theory. For any field \(k\) of characteristic 0 one can prove for algebraic cobordism the analogue of a theorem of Quillen on complex cobordism: the cobordism ring of the ground field is the Lazard ring \(\mathbb L\) and for any smooth \(k\)-variety \(X\), the algebraic cobordism ring \(\Omega^*(X)\) is generated, as an \(\mathbb L\)-module, by elements of nonnegative degree. This implies Rost's conjectured degree formula. One also gives a relation between the Chow ring, the \(K_0\) of a smooth \(k\)-variety \(X\) and \(\Omega^*(X)\). The technical construction of pullbacks is the subject of two talks. At the end one presents the state of advances on the conjectural isomorphism between the Levine-Morel construction of algebraic cobordism and the ``homotopical algebraic cobordism'', the cohomology theory represented by motivic Thom spectrum in the Morel-Voevodsky \(A^1\)-stable homotopy category. Contributions: -- Alexander Nenashev, Introduction to classical cobordism; p.885 -- Bernhard Hanke, Quillen's work on MU; p.888 -- Serge Yagunov, Oriented cohomology theories over a field; p.889 -- Jörg Schürmann, Survey of basic properties of algebraic cobordism; p.892 -- Franziska Heinloth, The construction of algebraic cobordism; p.895 -- Uwe Jannsen, Localization for algebraic cobordism; p.896 -- Ivan Panin, Homotopy invariance property and projective bundle theorem; p.899 -- Jel Riou, Universal property of K-theory; p.900 -- Annette Huber-Klawitter, \(\Omega^*(k)\) and the Lazard ring; p.902 -- Alexander Schmidt, Degree formulas; p.903 -- Christian Serpé, Cobordism and Chow groups; p.905 -- Jens Hornbostel, Steenrod operations and other degree formulas; p.906 -- Stefan Schröer, Some applications; p.908 -- Ania Otwinowska, Construction of pull-backs in algebraic cobordism. I; p.910 -- Jörg Wildeshaus, Construction of pull-backs in algebraic cobordism. II; p.911 -- Oliver Röndigs, The \(\mathbb A^1\)-homotopy approach to algebraic cobordism. I; p.912 --Joseph Ayoub, The motivic Thom spectrum \(M\mathbb G\ell\) and the algebraic cobordism \(\Omega^*(-)\); p.916