Editing Atiyah–Singer index theorem (section)

===K-theory===
Atiyah and Singer's first published proof used [[K-theory]] rather than cobordism. If ''i'' is any inclusion of compact manifolds from ''X'' to ''Y'', they defined a 'pushforward' operation ''i''<sub>!</sub> on elliptic operators of ''X'' to elliptic operators of ''Y'' that preserves the index. By taking ''Y'' to be some sphere that ''X'' embeds in, this reduces the index theorem to the case of spheres. If ''Y'' is a sphere and ''X'' is some point embedded in ''Y'', then any elliptic operator on ''Y'' is the image under ''i''<sub>!</sub> of some elliptic operator on the point. This reduces the index theorem to the case of a point, where it is trivial.