As the title indicates, this paper is concerned with the computation of the cohomology (with compact support) of generalized configuration spaces with arbitrary coefficients. For a space \(X\) and a partition \(T\) of \(\{1,\dots,n\}\), the author defines \[X(T) = \{ (x_1, \dots, x_n) \in X^n \mid x_i = x_j \iff i \text{ and } j \text{ are in the same block of } T \}.\] For an upwards closed subset \(U\) of the set of all partitions of \(\{1,\dots,n\}\), the generalized configuration space is then \(F(X,U) = X^n \setminus \bigcup_{T \in U} X(T)\). This definition includes the case of the usual configuration space of ordered points \(F(X,n) = \{ (x_1, \dots, x_n) \mid \forall i \neq j, \, x_i \neq x_j \}\).
The cohomology \(H^\bullet_c(F(X,n))\) depends on more than just the ring \(H^\bullet_c(X)\). In this paper, the author shows that \(H^\bullet_c(F(X,n))\) can be computed, as a representation of the symmetric group and with arbitrary coefficients (ring, sheaf, complex of sheaves) from the data of the \(\mathbb{E}_\infty\)-algebra structure on \(C^\bullet_c(X)\), which induces the ring structure of \(H^\bullet_c(X)\). In fact, the author proves even that \(H^\bullet_c(F(X,n))\) can be recovered from just the twisted commutative dg-algebra (tcdga) structure of \(C^\bullet_c(X)\), and that if one forgets the symmetric group action, \(H^\bullet_c(F(X,n))\) can be recovered from just the commutative shuffle dg-algebra structure of \(C^\bullet_c(X)\).
The proof uses a construction similar in spirit to one of M. Bendersky and S. Gitler [Trans. Am. Math. Soc. 326, No. 1, 423–440 (1991; Zbl 0738.54007)], who had produced a procedure to build a cdga \(\Lambda(n, A)\) from \(A = C^\bullet(X, \mathbb{Q})\) to compute \(H^\bullet(F(X,n), \mathbb{Q})\). The author vastly generalizes this construction to deal with tcdgas (to deal with coefficients different from \(\mathbb{Q}\)) and generalized configuration spaces. Given a set \(U\) as above and a tcdga, the author produces two cochain complexes \(\mathcal{CF}(U,A)\) and \(\mathcal{CD}(U, A)\). For a space \(X\) satisfying technical conditions and a bounded below complex of sheaves \(\mathcal{F}\) on \(X\), the author considers the tcdga \(R\Gamma^\otimes(X,\mathcal{F})\) (resp. \(R\Gamma^\otimes_c(X,\mathcal{F})\)) of derived global sections (resp. with compact support) of \(\mathcal{F}^{\otimes(-)}\). Then the author’s first theorem is that \(\mathcal{CF}(U, A)\) computes the cohomology (resp. with compact support) of \(F(X,U)\) with coefficients in \(\mathcal{F}^{\boxtimes n}\), and \(\mathcal{CD}(U,A)\) does the same for its complement in \(X^n\) (“generalized fat diagonal”). This permits the construction of several spectral sequences converging to the desired cohomologies.
The second main theorem of the paper is a generalization of a theorem of A. Arabia [“Espaces de configuration généralisés. Espaces topologiques \(i\)-acycliques. Suites spectrales “basiques””, Preprint, arXiv:1609.00522] regarding \(i\)-acyclic spaces, i.e. spaces \(X\) such that \(H^k_c(X) \to H^k(X)\) is the zero map for all \(k\). Here, the author proves that for such a space \(X\), the cdga \(C^\bullet_c(X,\mathbb{Q})\) is formal (i.e. quasi-isomorphic to its cohomology) and the cup product of its cohomology vanishes. By combining it with the first main theorem, this in particular recovers Arabia’s result that \(H^\bullet_c(F(X,n), \mathbb{Q})\) only depends on the graded vector space \(H^\bullet_c(X, \mathbb{Q})\). Like the first theorem, this result is generalized to arbitrary coefficients – with the caveat that \(H^\bullet_c(X,R)\) must be projective – and to generalized configuration spaces.