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Transactions of the American Mathematical Society

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$ e$-invariants and finite covers. II


Author: Larry Smith
Journal: Trans. Amer. Math. Soc. 347 (1995), 5009-5021
MSC: Primary 57R20; Secondary 55P42, 55Q45, 57M10
DOI: https://doi.org/10.1090/S0002-9947-1995-1316862-2
MathSciNet review: 1316862
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Abstract: Let $ \widetilde{M} \downarrow M$ be a finite covering of closed framed manifolds. By the Pontrijagin-Thom construction both $ \widetilde{M}$ and $ M$ define elements in the stable homotopy ring of spheres $ \pi _*^s$. Associated to $ \widetilde{M}$ and $ M$ are their $ e$invariants $ {e_L}(\widetilde{M})$, $ {e_L}(M) \in \mathbb{Q}/\mathbb{Z}$. If $ \widetilde{N} \downarrow N$ is a finite covering of closed oriented manifolds, then there is a related invariant $ {I_\Delta }(\widetilde{N} \downarrow N) \in \mathbb{Q}$ of the diffeomorphism class of the covering. In a previous paper we examined the relation between these invariants. We reduced the determination of $ {e_L}(\widetilde{M}) - p{e_L}(M)$, as well as $ {I_\Delta }(\widetilde{N} \downarrow N)$, for a $ p$-fold cover, to the evaluation of certain sums of roots of unity. In this sequel we show how the invariant theory of the cyclic group $ \mathbb{Z}/p$ may be used to evaluate these rums. For example we obtain

$\displaystyle \sum\limits_{\mathop {{\zeta ^p} = 1}\limits_{\zeta \ne 1} } {\fr... ... - 1}})}} {{(1 - \zeta )(1 - {\zeta ^{ - 1}})}} = \frac{{(p - 1)(p - 2)}} {3}} $

which may be used to determine the value of $ {I_\Delta }$ in degrees congruent to $ 3$ $ \mod 2(p - 1)$ for odd primes $ p$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1316862-2
Article copyright: © Copyright 1995 American Mathematical Society

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