##
**Betti numbers and stability for configuration spaces via factorization homology.**
*(English)*
Zbl 1377.57025

The main result of this paper is the construction of an algebraic model for the rational homology groups of the unordered configuration spaces of a manifold \(M\). Explicitly, the model is given as the Lie algebra homology of a Lie algebra built out of the compactly supported cohomology groups of \(M\). This Lie algebra is always either Abelian or globally nilpotent of order two, so the Lie algebra homology computation is often tractable. The construction has a secondary grading which tracks the number of points \(k\) in the configuration.

The paper contains other subsidiary results. It presents similar algebraic models for the homology of unordered configuration spaces with twisted coefficients and the homology of relative configuration spaces. There are also homological stability results sharpening statements elsewhere in the literature. Several examples are computed for configurations in the multiply punctured plane and various non-orientable surfaces.

Unordered configuration spaces have a rich literature, and the main and subsidiary results unify, streamline, and extend many extant results. For various classes of manifolds, including compact odd dimensional manifolds and nilpotent orientable compact even dimensional manifolds, the algebraic model presented here (or a closely related model) was already known to compute the homology of configuration spaces of \(M\). In these previous results it was necessary for technical reasons to perform a certain shift which meant that the models in question could not handle all \(k\) simultaneously. The paper under review eliminates these technicalities, treats all manifolds (compactness, orientability, and the other restrictions are no longer necessary), and handles every case on the same footing.

The tool used, factorization homology, is also new in this application. Factorization homology takes as input a certain kind of algebraic structure (an \(n\)-disk algebra \(A\)) and outputs an invariant of \(n\)-manifolds so that the value taken on \(\mathbb{R}^n\) is \(A\). The value on \(M\) is generated, very roughly, by gluing together copies of \(A\) along the charts of \(M\). Configuration spaces of manifolds have the same general kind of gluing structure, and so factorization homology turns out to be well-suited to studying them.

The main technical work here consists in identifying the relationship between two different (classes of) \(n\)-disk algebras: one class which makes the connections to configuration spaces manifest, and a second class related to Lie algebras which outputs the Lie homology-theoretic computational model. This is not only the technical heart of the paper, but also its ideological heart. That is, the paper’s results are intended as an advertisement for the concrete applicability of \(\infty\)-categorical homotopy theoretic tools (in this case, factorization homology) to classical problems elsewhere in mathematics (in this case, in algebraic topology).

Because of this, the paper is not altogether self-contained. It intimately uses various \(\infty\)-categorical results whose full exposition, elsewhere, constitute part of a large and growing body of foundational work in the theory. The seeming inextricability of the background material from the heart of the proof constitutes part of the same advertisement, arguing implicitly for the necessity of the full theory of \(\infty\)-categories.

The paper contains other subsidiary results. It presents similar algebraic models for the homology of unordered configuration spaces with twisted coefficients and the homology of relative configuration spaces. There are also homological stability results sharpening statements elsewhere in the literature. Several examples are computed for configurations in the multiply punctured plane and various non-orientable surfaces.

Unordered configuration spaces have a rich literature, and the main and subsidiary results unify, streamline, and extend many extant results. For various classes of manifolds, including compact odd dimensional manifolds and nilpotent orientable compact even dimensional manifolds, the algebraic model presented here (or a closely related model) was already known to compute the homology of configuration spaces of \(M\). In these previous results it was necessary for technical reasons to perform a certain shift which meant that the models in question could not handle all \(k\) simultaneously. The paper under review eliminates these technicalities, treats all manifolds (compactness, orientability, and the other restrictions are no longer necessary), and handles every case on the same footing.

The tool used, factorization homology, is also new in this application. Factorization homology takes as input a certain kind of algebraic structure (an \(n\)-disk algebra \(A\)) and outputs an invariant of \(n\)-manifolds so that the value taken on \(\mathbb{R}^n\) is \(A\). The value on \(M\) is generated, very roughly, by gluing together copies of \(A\) along the charts of \(M\). Configuration spaces of manifolds have the same general kind of gluing structure, and so factorization homology turns out to be well-suited to studying them.

The main technical work here consists in identifying the relationship between two different (classes of) \(n\)-disk algebras: one class which makes the connections to configuration spaces manifest, and a second class related to Lie algebras which outputs the Lie homology-theoretic computational model. This is not only the technical heart of the paper, but also its ideological heart. That is, the paper’s results are intended as an advertisement for the concrete applicability of \(\infty\)-categorical homotopy theoretic tools (in this case, factorization homology) to classical problems elsewhere in mathematics (in this case, in algebraic topology).

Because of this, the paper is not altogether self-contained. It intimately uses various \(\infty\)-categorical results whose full exposition, elsewhere, constitute part of a large and growing body of foundational work in the theory. The seeming inextricability of the background material from the heart of the proof constitutes part of the same advertisement, arguing implicitly for the necessity of the full theory of \(\infty\)-categories.

Reviewer: Gabriel C. Drummond-Cole (Gyeongbuk)

### MSC:

57R19 | Algebraic topology on manifolds and differential topology |

17B56 | Cohomology of Lie (super)algebras |

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

55N40 | Axioms for homology theory and uniqueness theorems in algebraic topology |

57N65 | Algebraic topology of manifolds |

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\textit{B. Knudsen}, Algebr. Geom. Topol. 17, No. 5, 3137--3187 (2017; Zbl 1377.57025)

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