The previous lemma implies that lower bounds for the Betti numbers of the homotopy totalization of a cosimplicial space [X.sup.*] can be obtained, when the H* BKSS collapses at the [E.sub.2]-page, by computing Euler characteristics at the level of the [E.sub.1]-page.
5 Euler Poincare series in the H* BKSS for spaces of links.
Let N [greater than or equal to] 4 and consider the H* BKSS of the Munson-Volic cosimplicial model for the space of long links with l strings..
Using Proposition 4.4, (5.8), Proposition 5.2, and (5.3), we compute that the Euler series of the first page of the H* BKSS is