On the fixed point indices and Nielsen numbers of fiber maps on Jiang spaces

Abstract:

Let $T = \{ E,P,B\}$ be a locally trivial fiber space, where E, B and ${P^{ - 1}}(b)$ for each $b \in B$ are compact, connected ANR’s (absolute neighborhood retracts). If $f:E \to E$ is a fiber (preserving) map then f induces $f’:B \to B$ and ${f_b}:{P^{ - 1}}(b) \to {P^{ - 1}}(b)$ for each $b \in B$ such that $Pf = f’P$. If E, B and ${P^{ - 1}}(b)$ for each $b \in B$ satisfy the Jiang condition then $N(f) \cdot P(T,f) = N(f’) \cdot N({f_b})$, and $i(f) = i(f’) \cdot i({f_b}) \cdot P(T,f)$ for each $b \in B$. If, in addition, the inclusion map $i:{P^{ - 1}}(b) \to E$ induces a monomorphism ${i_\# }:{\pi _1}({P^{ - 1}}(b)) \to {\pi _1}(E)$ and $f’$ induces a fixed point free homomorphism ${f’_\# }:{\pi _1}(B) \to {\pi _1}(B)$, then $N(f) = N(f’) \cdot N({f_b})$ and $i(f) = i(f’) \cdot i({f_b})$ for each $b \in B$. As an application, we prove: Let $T = \{ E,P,CP(n)\}$ be a principal torus bundle over an n-dimensional complex projective space $CP(n)$. If $f:E \to E$ is a fiber map such that for some $b \in CP(n),{f_b}:{P^{ - 1}}(b) \to {P^{ - 1}}(b)$ is homotopic to a fixed point free map, then there exists a map $g:E \to E$ homotopic to f and fixed point free.