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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Jingyal Pak
Journal: Trans. Amer. Math. Soc. 212 (1975), 403-415
MSC: Primary 55C20
MathSciNet review: 0420602
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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_\char93 }:{\pi _1}({P^{ - 1}}(b)) \to {\pi _1}(E)$ and $ f'$ induces a fixed point free homomorphism $ {f'_\char93 }:{\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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0420602-6
PII: S 0002-9947(1975)0420602-6
Keywords: Lefschetz number, Nielsen number, fixed point index, fiber map, complex projective space, aspherical manifold
Article copyright: © Copyright 1975 American Mathematical Society