The paper under review is a correction and extension of the author’s paper [“Vanishing powers of the Euler class”, Topology 40, No.5, 871–926 (2001; Zbl 0989.57016)] that contained a gap in the proof on p. 905. Now the vanishing result is redone under simpler hypotheses and, in addition, a non-vanishing result is included.
Let \(G_{\delta}\) be the discrete group of orientation preserving homeomorphisms of the circle \(S^ 1\), and let \(G_{\tau}\) denote the topological group of orientation preserving homeomorphisms of \(S^ 1\). As in contrast to \(G_{\tau}\), there are subgroups \(K\) of \(G_{\delta}\) for which the powers of the discrete Euler class are non-vanishing only up to a certain dimension. More precisely, the author distinguishes \(E_ K^ {n}\in H^ {2n}(BK; \mathbb Z)\) (with \(BK\) being the classifying space of \(K\)) from \(e_ K^ {n}\in \text{Hom}(H_ {2n}(BK; \mathbb Z),\mathbb Z)\), referring to \(e_ K^ {n}\) as the \(n\)th power of the discrete Euler class of \(K\), and to \(E_ K^ {n}\) as the \(n\)th power of the primary discrete Euler class of \(K\).
Supposing that we have a subgroup \(K \subset G_{\delta}\) acting on \(S^ 1\) by \(gx = g(x)\), this \(K\) acts on the infinite simplex \(\Delta^\infty\) which is a contractible simplicial complex whose \(q\)-simplices are \((q+1)\)-element subsets of points of \(S^ 1\). Supposing, in addition, that there exists a certain fundamental \(2p\)-simplex for the action of \(K\) with certain characteristic homeomorphisms that satisfy a certain commutativity condition and assuming that the isotropy subgroup for every \((2p-1)\)-simplex is trivial, the author proves the following main result: (a) \(E_ K^ {p}\) is an element of order \(2p(2p+1)\) in the cohomology group \(H^ {2p}(BK; \mathbb Z)\); (b) \(E_ K^ {n}=0\) for \(n>p\); (c) \(e_ K^ {n}=0\) for \(n\geq p\); (d) \(e_ K^ {n}\neq 0\) for \(n<p\). The author applies this result to the based mapping class group of a closed orientable surface of positive genus \(g\); he shows that one has \(p=g\) in this case.