Will anti-IgG secondary antibody cross-react with IgM primary antibody?

Secondary antibodies that react with IgG (H+L) will react with epitopes on both heavy and light chains, so they will react with other isotypes of primary antibody, such as IgM, or different subtypes of IgG (IgG2, IgG2a, etc.). Secondary antibodies that specify a specific isotype for their reactivity (such as Goat Anti-Mouse IgG2a, Isotype Specific) are cross-adsorbed against other isotypes for specific binding.