The theorem relates the Burnside ring of a finite group G to the stable cohomotopy of the classifying space BG.
The conjecture was made in the mid 1970s by Graeme Segal and proved in 1984 by Gunnar Carlsson.
On the right-hand side, the hat denotes the completion of the Burnside ring with respect to its augmentation ideal.
The answer is affirmative, and the representing object is called the classifying space of the group G and typically denoted BG.
It can be shown that if G is finite, then any CW-complex modelling BG has cells of arbitrarily large dimension.
The stable cohomotopy is in a sense the natural analog to complex K-theory, which is denoted
Segal was inspired to make his conjecture after Michael Atiyah proved the existence of an isomorphism which is a special case of the Atiyah–Segal completion theorem.