Mappings and homological properties in the Conley index theory

The role in the Conley index of mappings between flows is considered. A class of maps is introduced which induce maps on the index level. With the addition of such maps to the theory, the homology Conley index becomes a homology theory. Using this structure, an analogue of the Lefschetz theorem is proved for the Conley index. This gives a new condition for detecting fixed points of flows, extending the classical Euler characteristic condition.