Truncated symmetric products and configuration spaces (Q1323437)

There are two methods for studying the homology of the unordered configuration spaces \(C^ n(Y)\) of a manifold \(Y\). The method of [the third author with \textit{P. Loeffler}, Contemp. Math. 78, 415-424 (1988; Zbl 0667.55005)] uses the topological vector space \(TP^ \infty(Y)\) over \(\mathbb{F}_ 2\) generated by all points of \(Y\), and identifies \(C^ n(Y)\) essentially as the Poincaré dual of filtration quotients. The method of [the first two authors with \textit{L. Taylor}, Topology 28, 111-123 (1989; Zbl 0689.55012)] recognizes \(C^ n(Y)\) as a building block of the mapping space \(\text{map}(Y;S^ L)\), with \(L\) sufficiently large. Both methods determine the homology with coefficients in a field \(\mathbb{F} = \mathbb{Q}\), or \(\mathbb{F} = \mathbb{F}_ p\), as part of the homology of the larger spaces \(TP^ \infty(Y)\) or \(\text{map}(Y;S^ L)\). In the present article we give a parallel outline of both methods and their results. In addition, there are several new and related results about configuration spaces of surfaces and of highly connected manifolds, about alternating products, and about splittings of mapping spaces.