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Generalized Lefschetz numbers


Author: S. Y. Husseini
Journal: Trans. Amer. Math. Soc. 272 (1982), 247-274
MSC: Primary 55M20
DOI: https://doi.org/10.1090/S0002-9947-1982-0656489-X
MathSciNet review: 656489
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Abstract: Given $ [C;f]$, where $ C$ is a finitely-generated $ \pi $-projective chain complex, and $ f:C \to C{\text{a(}}\pi {\text{,}}\varphi {\text{)}}$-chain map, with $ \varphi :\pi \to \pi $ being a homomorphism, then the generalized Lefschetz number $ {L_{(\pi ,\varphi )}}[C;f]$ of $ [C;f]$ is defined as the alternating sum of the $ (\pi ,\varphi )$-Reidemeister trace of $ f$. In analogy with the ordinary Lefschetz number, $ {L_{(\pi ,\varphi )}}[C;f]$ is shown to satisfy the commutative property and to be invariant under $ (\pi ,\varphi )$-chain homotopy. Also, when $ {H_\ast}C$ is $ \pi $-projective,

$\displaystyle {L_{(\pi ,\varphi )}}[C;f] = {L_{(\pi ,\varphi )}}[{H_\ast}C;{H_\ast}f]$

If $ \pi ' \subset \pi $ is $ \varphi $-invariant and with finite index, then for $ \alpha \in \pi '$, the $ (\pi ',\varphi )$Reidemeister class $ [\alpha ;\pi ']$ is essential for $ f:C \to C$ if and only if $ {[\alpha ;\pi ]_\varphi }$ is essential. If $ \pi ' \subset \pi $ is normal, then one can use the cosets of $ \pi \operatorname{mod} \pi '$ to detect the essential $ (\pi ,\varphi )$-classes of $ f:C \to C$. This is expressed as a decomposition of $ {L_{(\pi ,\varphi )}}[C;f]$ in terms of $ {L_{(\pi ',{\varphi _\xi })}}[C';{f_\xi }]$ where $ f( \cdot ){\xi ^{ - 1}} = {f_\xi }( \cdot )$ and $ {\varphi _\xi }( \cdot ) = \xi \varphi ( \cdot ){\xi ^{ - 1}}$. The algebraic theory is applied to the Nielsen theory of a map $ f:X \to X$, where $ X$ is a finite CW-complex relative to a regular cover $ \tilde X \to X$. One can define a generalized Lefschetz number $ {L_{(\pi ,\varphi )}}$ using any cellular approximation to $ f$, where $ \pi $ is the group of covering transformations of $ \tilde X \to X$. The quantity $ {L_{(\pi ,\varphi )}}$ can be expressed naturally as a formal sum in the $ \pi $-Nielsen classes of $ f$ with their indices appearing as coefficients. From this expression, one is able to deduce from the properties of the generalized Lefschetz number the usual results of the relative Nielsen theory.

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9947-1982-0656489-X
Article copyright: © Copyright 1982 American Mathematical Society

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