Homotopy and Cohomology
Funded by the DFG (Deutsche Forschungsgemeinschaft).
Mrs. Sabine George
Endenicher Allee 60, Room N1.003 (Neubau/Annexe)
53115 Bonn, Germany
Tel.: +49 228 73-2510
Fax: +49 228 73-6490
Office hours: Thursday 9-15
Homotopy and Cohomology
Topology stands out amongst other branches of mathematics for the way it bridges the gap between the realm of continuous phenomena (geometry and analysis) and the discrete world (algebra and combinatorics). Topology uses discrete techniques to study continuous objects; it has assimilated methods from many areas of mathematics, and methods developed by topologists have in turn contributed significantly to advances in other areas.
Topology is concerned with the study of geometric objects (such as manifolds) and of the continuous maps between them. Typically one asks whether there are any geometric objects having certain specified properties; and if so, how many different objects share these properties. The technical term is that one seeks a classification of such objects, or of the maps between them. One example would be classifying the covering spaces of a given manifold.
Many of the properties one is interested in are retained when the object under investigation is subjected to a deformation. One example is the degree of a mapping: the degree (an integer) is unchanged when the mapping is continuously deformed. One says that the degree is invariant up to homotopy. The power of Homotopy Theory lies in this invariance up to homotopy: often it allows one to replace complicated objects by simple models of them. The strategy of Homotopy Theory is to establish this invariance for as many properties as possible, and then exploit the invariance to obtain a classification.
Classifying objects usually involves calculating homotopy groups. As the direct approach is typically prohibitively difficult, one uses cohomology to calculate these homotopy groups by indirect means. That is, one makes use of both ordinary (i.e. singular) cohomology and generalized cohomology theories, together with the associated ring structure and cohomology operations.
Recent years have seen several new and promising developments leading to even greater interconnections between Homotopy Theory on the one hand and Algebra, Algebraic Geometry and Theoretical Physics on the other. The areas of Homotopy Theory in question are Stable Homotopy Theory and Elliptic Cohomology.
These new theories and methods will provide the thesis topics in the proposed Research Training Group. To be more specific, we will concentrate on the following themes: Classifying spaces and cohomology of groups, Configuration spaces and mapping spaces, Moduli spaces, Stable homotopy theory, Elliptic cohomology and topological modular forms, Operads and E-infinity structures, Manifolds and bordism theory, Cohomology and homotopy theory of foliations.
Last modified: 11.03.2014