The degree of maps of free G-manifolds and the average |
| |
Authors: | Jan Jaworowski |
| |
Institution: | (1) Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA |
| |
Abstract: | Suppose that G is a compact Lie group, M and N are orientable, free G-manifolds and f : M → N is an equivariant map. We show that the degree of f satisfies a formula involving data given by the classifying maps of the orbit spaces M/G and N/G. In particular, if the generator of the top dimensional cohomology of M/G with integer coefficients is in the image of the cohomology map induced by the classifying map for M, then the degree is one.
The condition that the map be equivariant can be relaxed: it is enough to require that it be “nearly equivariant”, up to a
positive constant. We will also discuss the G-average construction and show that the requirement that the map be equivariant can be replaced by a somewhat weaker condition
involving the average of the map.
These results are applied to maps into real, complex and quaternionic Stiefel manifolds. In particular, we show that a nearly
equivariant map of a complex or quaternionic Stiefel manifold into itself has degree one.
Dedicated to the memory of Jean Leray |
| |
Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 5515 Secondary 55R40 55M25 57R91 |
本文献已被 SpringerLink 等数据库收录! |
|