The rational homotopy type of the complement of the graph of a map (Q505022)

The authors study the rational homotopy type of the complement \(C(f)\) of the graph of a continuous map \(f: M\to N\) from a simply connected closed manifold to a 2-connected closed manifold of the same dimension. If \(f\) is the identity map, this corresponds to the configuration space of two points studied by \textit{P. Lambrechts} and \textit{D. Stanley} [Ann. Inst. Fourier 54, No. 4, 1029--1052 (2004; Zbl 1069.55006)]. Here, the authors extend a result of [loc. cit.] by proving that the rational homotopy type of \(C(f)\) depends only on the rational homotopy type of \(f\). They also describe a Sullivan model of \(C(f)\) which is a relative version of the Lambrechts-Stanley model. In particular, they prove that \(C(f)\) is a formal space if the map \(f\) is formalisable. An example shows that the converse is not true, in opposition with the Lambrechts-Stanley result on the configuration space of two points.