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The generalized algebraic conjecture on spherical classes

  • Nguyễn H. V. HưngEmail author
  • Ngô A. Tuấn
Article
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Abstract

Let X be a pointed CW-complex. The generalized conjecture on spherical classes states that, the Hurewicz homomorphism\(H: \pi _{*}(Q_0 X) \rightarrow H_{*}(Q_0 X)\)vanishes on classes of\(\pi _* (Q_0 X)\)of Adams filtration greater than 2. Let \( \varphi _s: {\rm Ext} _{\mathcal {A}}^{s}(\widetilde{H}^*(X), \mathbb {F}_2) \rightarrow {(\mathbb {F}_2 \otimes _{{\mathcal {A}}}R_s\widetilde{H}^*(X))}^* \) denote the sth Lannes–Zarati homomorphism for the unstable \({\mathcal {A}}\)-module \(\widetilde{H}^*(X)\). This homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the sth Lannes–Zarati homomorphism vanishes in any positive stem for \(s>2\) and any CW-complex X. We construct a chain level representation for the Lannes–Zarati homomorphism by means of modular invariant theory. We show the commutativity of the Lannes–Zarati homomorphism and the squaring operation. The second Lannes–Zarati homomorphism for \(\mathbb {R}\mathbb {P}^{\infty }\) vanishes in positive stems, while the first Lannes-Zatati homomorphism for any space is basically non-zero. We prove the algebraic conjecture for \(\mathbb {R}\mathbb {P}^{\infty }\) and \(\mathbb {R}\mathbb {P}^{n}\) with \(s=3\), 4. We discuss the relation between the Lannes–Zarati homomorphisms for \(\mathbb {R}\mathbb {P}^{\infty }\) and \(S^0\). Consequently, the algebraic conjecture for \(X=S^0\) is re-proved with \(s=3\), 4, 5.

Mathematics Subject Classification

Primary 55P47 55Q45 55S10 55T15 

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Notes

Acknowledgements

This work was in progress when the authors recently visited the Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi. They would like to express their warmest thanks to the VIASM for the hospitality and for the wonderful working condition.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityHanoiVietnam

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