The Euler class in homological algebra
The topological group \(G_\tau\) of orientation preserving homeomorphisms of the circle is homotopy equivalent to \(S^1 = U(1)\). It is not abelian however, but simple, and the identity \(G_\delta \to G_\tau\) from the same group equipped with the discrete topology induces a homology equivalence \(BG_\delta \to BG_\tau \simeq \mathbb C P^\infty\). The author provides a computational proof of this fact (due to Mather and generalized by Thurston) based on bisimplicial models. He also constructs a group extension \(0 \to \mathbb Z \to G_{-} \to G_\delta \to 1\), which he calls ``the discrete Euler Class'' of \(G\) since it corresponds via the above homology equivalence to the usual, topological, Euler Class. The value of this Euler Class is shown to be bounded on any space.
scientific article; zbMATH DE number 5960892
scientific article