Document Zbl 0666.55008 - zbMATH Open
Rational cohomology of configuration spaces of surfaces. (English) Zbl 0666.55008
Algebraic topology and transformation groups, Proc. Conf., Göttingen/FRG 1987, Lect. Notes Math. 1361, 7-13 (1988).
[For the entire collection see Zbl 0646.00011.]
Consider \(\tilde C^ k(M)=\{space\) of all ordered k-tuples of distinct points of \(M\}\) and the configuration space \(C^ k(M)=\tilde C^ k(M)/\Sigma_ K\) for a given manifold M. The paper under review deals with the (additive) computation of \(H*(C^ k(M);{\mathbb{Q}})\), for M \(=\) closed oriented surface of genus g, \(M_ g\), minus a point. Main result: Defining, for any g and n, a free differential graded commutative algebra \({\mathcal M}(g,n)\) by \({\mathcal M}(g,n)=(\Lambda (z_ 1,...,z_{2g},v,u_ 1,...,u_{2g},w),d)\), where \(\deg (z_ i)=2n+1,\) \(\deg (v)=2n\), \(\deg (u_{ij})=4n+2,\) \(\deg (w)=4n+1,\) and where the differential d acts trivially on the generators except \(dw=2(z_ 1z_ 2+...+z_{2g- 1}z_{2g}),\) and bigrading it by further assigning the weight 1 to v and \(z_ i\) and the weight 2 to w and \(u_ i\), then, given g and k, there exists a positive integer \(n=n(g,k)\) such that \(H*(C^ k(M);{\mathbb{Q}})\) is isomorphic as a graded vector space to \(H*_{weight}=k^{{\mathcal M}(g,n)}\) desuspended 2nk times. A complete computation of \(H^*_*{\mathcal M}(g,n)\) is also included. The proof is first reduced to the computation of the cohomology of a stable construction denoted by \(C(M_ g\setminus \{point\};S^{2n})\), which is known to be equivalent to a function space of based maps, namely to \(map_ 0(M_ g,S^{2n+2})\); second the (additive) homology computation for the function space is achieved by means of a Serre spectral sequence. It is to be noted that a slight modification of a method due to Haefliger [see, e.g., J. M. Möller and M. Raußen, Trans. Am. Math. Soc. 292, 721-732 (1985; Zbl 0605.55008)] should give in fact more, namely that dga \({\mathcal M}(g,n)\) represents the \({\mathbb{Q}}\)-homotopy type of \(map_ 0(M_ g,S^{2n+2})\).