Kobordismentheorie

These notes were taken from lectures given by tom Dieck in the win ter-term 1969/70 at the Mathematical Institute in Heidelberg. The aim of the lectures was to introduce the students to cobordism theory and to propagate ideas of Boardman and Quillen about the calculation of cobordism theories with the aid of formal groups. These notes give an enlarged version of the leetures with many details and proofs filled in. A chapter on unitary cobordism has been left out and will appear separately. The eontents of the notes are as follows: In chapter I we recall those parts of differential topology and of the theory of veetor bundles which we will use. This~ only to re wind the reader of well known faets or to give hints at neeessary pre requisites to students willing to learn differential topology. Apart from these faets we assume knowledge of elementary homotopy theory and classical cohomology with coefficients in l2 , characterized by the Eilenberg-Steenrod axioms. In chapter II the (non oriented) bordism homology theory N.(-) is defined by singular manifolds. We verify the axioms of a homology theory. Our approach differs from that of Conner and Floyd [4] in that we only define absolute homology groups and use a system of axioms in which an exact sequence of Mayer-Vietoris type plays the main role.

Klappentext

These notes were taken from lectures given by tom Dieck in the win­ ter-term 1969/70 at the Mathematical Institute in Heidelberg. The aim of the lectures was to introduce the students to cobordism theory and to propagate ideas of Boardman and Quillen about the calculation of cobordism theories with the aid of formal groups. These notes give an enlarged version of the leetures with many details and proofs filled in. A chapter on unitary cobordism has been left out and will appear separately. The eontents of the notes are as follows: In chapter I we recall those parts of differential topology and of the theory of veetor bundles which we will use. This~ only to re­ wind the reader of well known faets or to give hints at neeessary pre­ requisites to students willing to learn differential topology. Apart from these faets we assume knowledge of elementary homotopy theory and classical cohomology with coefficients in l2 , characterized by the Eilenberg-Steenrod axioms. In chapter II the (non oriented) bordism homology theory N.(-) is defined by singular manifolds. We verify the axioms of a homology theory. Our approach differs from that of Conner and Floyd [4] in that we only define absolute homology groups and use a system of axioms in which an exact sequence of Mayer-Vietoris type plays the main role.

Inhalt
I. Kapitel: Vorbereitungen.- II. Kapitel: Die Bordismen-Homologie-Theorie.- III. Kapitel: Darstellung von Bordismengruppen als Homotopiegruppen.- IV. Kapitel: Spektren, Homologie und Kohomologie.- V. Kapitel: Verträglichkeit der Kohomologie mit dem Limes.- VI. Kapitel: Charakteristische Klassen.- VII. Kapitel: Formale Gruppen.- VIII. Kapitel: Multiplikative Transformationen.- IX. Kapitel: Steenrod-Operationen in der Kobordismentheorie.- X. Kapitel: Charakteristische Zahlen.- XI. Kapitel: Stabile Operationen.- XII. Kapitel: Bordismus und Kobordismus.