This stimulating book (written in french) is an extended version of a series of online lectures given by Nicolas Bergeron in 2020 for the “Centre de Recherches Mathématiques” (CRM) in Montreal. It elaborates on constructions of “Eisenstein cocycles” for arithmetic subgroups of the general linear group developed by Sczech and many others. The authors devise a beautiful general construction of cocycles based on a topological insight. The introduction and the first two chapters of the book present the main results and explain the core construction. The following chapters discuss the details and initiate an investigation of the constructed cocycles. It seems expedite to follow this structure in the review.
The introduction nicely explains the objectives of the book starting from the classical addition formula for the cotangent function. Eisenstein’s approach to this formula via the function \(\varepsilon(x) = \frac{1}{2i} \cot(\pi x)\) is reviewed and it is explained how the addition formula can be related to an explicit \(\overline{\mathcal{M}}(\mathbb{C}^2/\mathbb{Z}^2)\)-valued modular symbol and thus to a 1-cocycle for \(\mathrm{SL}_2(\mathbb{Z})\); here \(\overline{\mathcal{M}}\) denotes meromorphic functions modulo constant functions. It is possible to describe the action of the Hecke operators on the modular symbol and moreover, it is indicated how cotangent functions can be used to give explicit \(\mathcal{M}(\mathbb{C}^2/\mathbb{Z}^2)\)-valued cocycles for congruence subgroups \(\Gamma_0(N)\) in \(\mathrm{SL}_2(\mathbb{Z})\). A theorem describing the Hecke action on the associated cohomology classes is presented. Then the book sets off to develop a far reaching generalization of this blueprint result.
The first chapter presents the topological construction that gives rise to the cocycles investigated throughout the book. The construction starts with a group \(G\), a \(G\)-space \(X\) and a finite \(G\)-stable subset \(F \subseteq X\). Throughout the authors investigate the following three cases:

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affine: \(X = \mathbb{C}^n\) and \(F = \{0\}\) with \(G = \mathrm{GL}_n(\mathbb{C})\) considered as a discrete group.

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mulitplicative: \(X = \mathbb{C}^n/\mathbb{Z}^n\) and \(F\) is a set of torsion points in \(X\) where \(G\) a finite index subgroup of \(\mathrm{GL}_n(\mathbb{Z})\).

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elliptic: \(X = E^n\) for an elliptic curve \(E\) and \(F\) is a set of torsion points in \(X\) where \(G\) is a finite index subgroup of \(\mathrm{GL}_n(\mathbb{Z})\) (or \(\mathrm{GL}_n(\mathcal{O})\) if \(E\) admits multiplication by \(\mathcal{O}\).)

The Thom isomorphism is used to construct a class \([D]\) in the equivariant cohomology \(H_G^{2n}(X,X\setminus F)\) for suitable \(D \in H_G^0(F)\). It is shown later in the book that in all three cases \([D]\) admits a distinguished lift \(E[D]\) to \(H_{G}^{2n-1}(X\setminus F)\). In the affine case a theorem of Brieskorn can be used to promote \(E[D]\) to a class \(S_{\text{aff}}[D] \in H^{n-1}(G,\Omega_{\text{mer}}^n(\mathbb{C}^n))\) where \(\Omega_{\text{mer}}^n\) denotes the space of meromorphic \(n\)-forms. Similarly, in the multiplicative case the authors obtain a class \(S_{\text{mult}}[D] \in H^{n-1}(G,\Omega_{\text{mer}}^n(\mathbb{C}^n/\mathbb{Z}^n))\) and in the elliptic case a class \(S_{\text{ell}}[D] \in H^{n-1}(G,\Omega_{\text{mer}}^n(E^n))\). A suitable substitute for the theorem of Brieskorn for the multiplicative and elliptic case is proven in Chapter 3 (Theorem 3.5).
The second chapter states the main results. In the affine case two explicit cocycles \(S_{\text{aff}}, S_{\text{aff}}^*\) are described, that represent the cohomology class constructed in Chapter 1. The construction of \(S_{\text{aff}}^*\) hinges on a new relation between the Steinberg module and the algebra of “affine” \(n\)-form \(\Omega^n_{\text{aff}}\) that translates the Ash-Rudolph relations for universal modular symbols to the Orlik-Solomon relations.
Similarly, in the multiplicative case the authors describe cocycles that represent the cohomology class \(S_{\text{mult}}[D] \in H^{n-1}(\Gamma,\Omega_{\text{mer}}^n(\mathbb{C}^n/\mathbb{Z}^n))\) constructed before, where \(\Gamma\) is a finite index subgroup of \(\mathrm{SL}_n(\mathbb{Z})\). As in the special case \(n=2\), they are able to describe Hecke operators on the class (Theorem 2.3 is proven in Chapter 8). In addition, the class is represented by partial modular symbols valued in \(\mathcal{M}(\mathbb{C}^n/\mathbb{Z}^n)\) that can be described explicitly using \(n\)-fold products of cotangent functions. By evaluating the cocycles at 0 the authors obtain non-trivial classes in \(H^{n-1}(\Gamma_0(N),\mathbb{Q})\) with information on the denominator and the Hecke action.
The elliptic case is the most involved and the proofs are discussed in chapter 9. Again representatives for the class \(S_{\text{ell}}[D]\) are constructed, the Hecke action is described and associated \(\mathcal{M}(E^n)\)-valued partial modular symbols can be described using \(n\)-fold products of the Eisenstein series \(E_1\).
Chapter 3 discusses the replacement for the theorem of Brieskorn in the non-affine cases. Chapter 4 explains the construction of two differential forms on \(S \times \mathbb{C}^n\) where \(S\) denotes the symmetric space for \(\mathrm{GL}_n(\mathbb{C})\) that are essential for the construction. Chapter 5 is concerned with the behavior of the constructed forms close to the boundary of the associated symmetric spaces and Chapter 7 studies Eisenstein series involving the differential forms. Chapter 6 deal with the details for the affine case, Chapter 8 with the multiplicative case and Chapter 9 with the elliptic case. The appendix contains a very brief review of equivariant cohomology.
The book opens an intriguing perspective on “Eisenstein cocycles” and will surely inspire future work in this area. Videos of the original lectures by Bergeron are available in the YouTube channel of CRM.