Index group

The group G0 consists of elements homotopic, in G, to the identity in G, the constant function 1.

One can choose the functions fn(z) = zn as representatives in G of distinct homotopy classes of maps T→T.

By Atkinson's theorem, an invertible elements in K is of the form π(T) where T is a Fredholm operators.

In fact, ΛK is isomorphic to the additive group of integers Z, via the Fredholm index.

In other words, for Fredholm operators, the two notions of index coincide.