Segal conjecture

Segal's Burnside ring conjecture, or, more briefly, the Segal conjecture, is a theorem in homotopy theory, a branch of mathematics. 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. As of 2016^[update], this statement is still commonly referred to as the Segal conjecture, even though it now has the status of a theorem.

The Segal conjecture has several different formulations, not all of which are equivalent. Here is a weak form: there exists, for every finite group G, an isomorphism

Here, lim denotes the inverse limit, π_S* denotes the stable cohomotopy ring, B denotes the classifying space, the superscript k denotes the k-skeleton, and the subscript + denotes the addition of a disjoint basepoint. On the right-hand side, the hat denotes the completion of the Burnside ring with respect to its augmentation ideal.

The Burnside ring of a finite group G is constructed from the category of finite G-sets as a Grothendieck group. More precisely, let M(G) be the commutative monoid of isomorphism classes of finite G-sets, with addition the disjoint union of G-sets and identity element the empty set (which is a G-set in a unique way). Then A(G), the Grothendieck group of M(G), is an abelian group. It is in fact a free abelian group with basis elements represented by the G-sets G/H, where H varies over the subgroups of G. (Note that H is not assumed here to be a normal subgroup of G, for while G/H is not a group in this case, it is still a G-set.) The ring structure on A(G) is induced by the direct product of G-sets; the multiplicative identity is the (isomorphism class of any) one-point set, which becomes a G-set in a unique way.

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