The Topology of $SU(\mathrm{\infty })$ and the Group of Area-Preserving Diffeomorphisms of a Compact 2-manifold

Given the interest in relating the large $N$ limit of SU(N) to groups of area-preserving diffeomorphisms, we consider the topologies of these groups and show that both in terms of homology and homotopy, they are extremely different. Similar conclusions are drawn for other infinite dimensional classical groups.