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**Real homotopy of configuration spaces. Peccot lecture, Collège de France, March & May 2020.**
*(English)*
Zbl 1496.55001

Lecture Notes in Mathematics 2303. Cham: Springer; Lille: Centre Européen pour les Mathématiques, la Physiques et leurs Interactions (CEMPI) (ISBN 978-3-031-04427-4/pbk; 978-3-031-04428-1/ebook). xviii, 185 p. (2022).

The notion of the configuration space \(\operatorname{Conf}_{r}(M)\) of ordered \(r\)-tuples of distinct points in a manifold \(M\) has played an important role in algebraic geometry and topology for over a century, though their homotopy theory was first studied in an orderly way by E. Fadell and L. Neuwirth [Math. Scand. 10, 111–118 (1962; Zbl 0136.44104)]. In particular, these spaces provide important invariants of manifolds (see E. R. Fadell and S. Y. Husseini’s survey [Geometry and topology of configuration spaces. Berlin: Springer (2001; Zbl 0962.55001)]). However, an example due to R. Longoni and P. Salvatore [Topology 44, No. 2, 375–380 (2005; Zbl 1063.55015)] shows that configuration spaces are not a homotopy invariant, even for closed manifolds.

The present monograph, on the real homotopy type of configuration spaces, is based on the Peccot Lectures given by the author at the Collège de France in the spring of 2020. P. Lambrechts and D. Stanley [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 497–511 (2008; Zbl 1172.13009)] constructed a certain Poincaré duality commutative differential graded algebra (CDGA) \(G_{A}(r)\) from a CDGA model \(A\) for a simply-connected closed manifold \(M\), and in his thesis, published in [Invent. Math. 216, No. 1, 1–68 (2019; Zbl 1422.55031)], the author showed that this \(G_{A}(r)\) is indeed a real CDGA model for \(\operatorname{Conf}_{r}(M)\), as they had conjectured. In [“A model for configuration spaces of points”, Preprint, arXiv:1604.02043], R. Campos and T. Willwacher constructed an alternative real CDGA model for \(\operatorname{Conf}_{r}(M)\), using certain graph complexes based on Kontsevich’s graph cooperad, and used this for an alternative proof of the real homotopy invariance (described in detail in Chapter 3 of the present book). These ideas were then used, in collaboration with Idrissi and Lambrechts, to prove a version of this statement for manifolds with boundary (described in Chapter 4). Finally, Chapter 5 elucidates the relationship between configuration spaces, factorization homology, operads, and formality.

The present monograph, on the real homotopy type of configuration spaces, is based on the Peccot Lectures given by the author at the Collège de France in the spring of 2020. P. Lambrechts and D. Stanley [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 497–511 (2008; Zbl 1172.13009)] constructed a certain Poincaré duality commutative differential graded algebra (CDGA) \(G_{A}(r)\) from a CDGA model \(A\) for a simply-connected closed manifold \(M\), and in his thesis, published in [Invent. Math. 216, No. 1, 1–68 (2019; Zbl 1422.55031)], the author showed that this \(G_{A}(r)\) is indeed a real CDGA model for \(\operatorname{Conf}_{r}(M)\), as they had conjectured. In [“A model for configuration spaces of points”, Preprint, arXiv:1604.02043], R. Campos and T. Willwacher constructed an alternative real CDGA model for \(\operatorname{Conf}_{r}(M)\), using certain graph complexes based on Kontsevich’s graph cooperad, and used this for an alternative proof of the real homotopy invariance (described in detail in Chapter 3 of the present book). These ideas were then used, in collaboration with Idrissi and Lambrechts, to prove a version of this statement for manifolds with boundary (described in Chapter 4). Finally, Chapter 5 elucidates the relationship between configuration spaces, factorization homology, operads, and formality.

Reviewer: David Blanc (Haifa)

### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55R80 | Discriminantal varieties and configuration spaces in algebraic topology |

55P62 | Rational homotopy theory |

18M75 | Topological and simplicial operads |

57R19 | Algebraic topology on manifolds and differential topology |