Let \[ \mathrm{Conf}_k(X)=\{ (x_1,\ldots, x_k)\in X^k\mid x_i\not= x_j\text{ if }i\not= j\} \] denote the ordered configuration space of \(k\) points in a topological space \(X\). In this paper, the authors study the homology of an ordered configuration space of a product \(X=M\times N\), where \(M\) and \(N\) are parallelizable manifolds of dimensions \(m\) and \(n\), respectively. Building on earlier work [W. Dwyer et al., Trans. Am. Math. Soc. 371, No. 4, 2963–2985 (2019; Zbl 1408.18016)], they consider the sequence \(\mathrm{Conf}(M)=(\mathrm{Conf}_k(M))_{k\geq 0}\) of configuration spaces of \(M\) as a module over the little \(m\)-cubes operad \({\mathcal{C}}_m\), and similarly for \(N\) and \(M\times N\).
Let \(R\) be a commutative ring, and assume \(H_\ast(\mathrm{Conf}_k(M);R)\) and \(H_\ast(\mathrm{Conf}_k(N);R)\) are \(R\)-projective for all \(k\geq 0\). The main theorem of the paper says that there is a natural, convergent spectral sequence \[ E^2_{p,q}\cong H_p\bigl( H_\ast(\mathrm{Conf}(M);R)\bigstar^{\mathbb{L}} H_\ast(\mathrm{Conf}(N);R)\bigr)_q \Rightarrow H_{p+q}(\mathrm{Conf}(M\times N);R) \] of \(R\)-linear \({\mathcal{C}}_{m+n}\)-modules. Here \(\bigstar\) denotes the linearized Boardman-Vogt tensor product. The authors also show that if \(R\) is a field of characteristic zero, then this spectral sequence degenerates at the term \(E^2\). The proof of the collapse relies on Kontsevich’s formality theorem, as extended by B. Fresse and T. Willwacher [J. Eur. Math. Soc. (JEMS) 22, No. 7, 2047–2133 (2020; Zbl 1445.18014)], and applies to any field over which formality holds.
These results reduce the computation of the rational homology of ordered configuration spaces of products – assuming knowledge of the factors – to a purely algebraic problem in the representation theory of certain combinatorial categories