Rational model for the string coproduct of pure manifolds
For a closed oriented manifold \(M\), the homology loop product is a product on the homology of its free loop space $LM=\mathrm{Maps}(S^1,M)$ defined as the family of products, $H_p(LM)\otimes H_q(LM)\longrightarrow H_{p+q-n}(LM)$, where $H_*$ denotes singular homology with integral coefficients and $n=\dim M$. In [Lond. Math. Soc. Lect. Note Ser. 308, 344--357 (2004; Zbl 1088.81082)], \textit{D. Sullivan} introduced a coproduct on the relative homology: $\vee:H_*(LM,M)\longrightarrow H_*(LM,M)\times (LM,M))$, induced by cutting loops, called here the string coproduct, and proved that the string coproduct and the loop product give an infinitesimal bialgebra structure when the Euler characteristic of $M$ is zero. In [``On the invariance of the string topology coproduct'', Preprint, \url{arXiv:1908.03857}], \textit{N. Hingston} and \textit{N. Wahl} proved that the coproduct is homotopy invariant, likewise for the loop product. They also lifted the coproduct to the chain level and likewise the non-relative coproduct in [``Product and coproduct in string topology'', Preprint, \url{arXiv:1709.06839}]. The aim of this paper is to study the string coproduct in rational homotopy theory. In Theorem 5.4.1, the author gives a Sullivan model for the dual to the string coproduct $H_*(LM,M)\otimes H_*(LM,M)\longrightarrow H_*(LM,M)$, when $M$ is a pure manifold. For pure manifolds, Theorem 1.0.1 shows that the degree with respect to the Hodge decomposition is determined by homotopy theoretic information. The behavior for non-pure manifolds is mostly unknown.
scientific article; zbMATH DE number 7454725
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