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String Topology, Euler Class and TNCZ free loop fibrations

Abstract : Let $M$ be a connected, closed oriented manifold. Let $\omega\in H^m(M)$ be its orientation class. Let $\chi(M)$ be its Euler characteristic. Consider the free loop fibration $\Omega M\buildrel{i}\over\hookrightarrow LM\buildrel{ev}\over\twoheadrightarrow M$. For any class $a\in H^*(LM)$ of positive degree, we prove that the cup product $\chi(M)a\cup ev^*(\omega)$ is null. In particular, if $i^*:H^*(LM;\mathbb{F}_p)\twoheadrightarrow H^*(\Omega M;\mathbb{F}_p)$ is onto then $\chi(M)$ is divisible by $p$ (or $M$ is a point).
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https://hal.archives-ouvertes.fr/hal-00855785
Contributor : Luc Menichi <>
Submitted on : Friday, August 30, 2013 - 9:39:46 AM
Last modification on : Monday, March 9, 2020 - 6:16:02 PM
Document(s) archivé(s) le : Thursday, April 6, 2017 - 10:39:21 AM

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Loop_coproduct_generalise.pdf
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  • HAL Id : hal-00855785, version 1
  • ARXIV : 1308.6684

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Luc Menichi. String Topology, Euler Class and TNCZ free loop fibrations. 2013. ⟨hal-00855785⟩

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