Product formula for the fixed point index of a fibre preserving map defined locally

Abstract

There is well known product formula for the index of a fixed point class of a fibre preserving map which shows that the index in the total space is the product of indices in the fibre and base. In this paper we generalize these ideas to the setting of fibre preserving maps defined locally. Let (E,p,B) be a fibre space with E,B and all fibres compact connected ANR's, V be an open subset in B, and U=p−1(V). Let f:U→E be a fibre preserving map, i.e., a map for which there exists a map f¯:V→B with pf=f¯pU,V, where pU,V:U→V is the restriction of p. We study the structure of the fixed point classes of f, and prove product formula for the index of a fixed point class of f.