Segal's conjecture

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

${\displaystyle \varprojlim \pi _{S}^{0}\left(BG_{+}^{(k)}\right)\to {\widehat {A}}(G).}$

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.

The Burnside ring is the analogue of the representation ring in the category of finite sets, as opposed to the category of finite-dimensional vector spaces over a field (see motivation below). It has proven to be an important tool in the representation theory of finite groups.

For any topological group G admitting the structure of a CW-complex, one may consider the category of principal G-bundles. One can define a functor from the category of CW-complexes to the category of sets by assigning to each CW-complex X the set of principal G-bundles on X. This functor descends to a functor on the homotopy category of CW-complexes, and it is natural to ask whether the functor so obtained is representable. The answer is affirmative, and the representing object is called the classifying space of the group G and typically denoted BG. If we restrict our attention to the homotopy category of CW-complexes, then BG is unique. Any CW-complex that is homotopy equivalent to BG is called a model for BG.

For example, if G is the group of order 2, then a model for BG is infinite-dimensional real projective space. It can be shown that if G is finite, then any CW-complex modelling BG has cells of arbitrarily large dimension. On the other hand, if G = Z, the integers, then the classifying space BG is homotopy equivalent to the circle S¹.

The content of the theorem becomes somewhat clearer if it is placed in its historical context. In the theory of representations of finite groups, one can form an object ${\displaystyle R[G]}$ called the representation ring of ${\displaystyle G}$ in a way entirely analogous to the construction of the Burnside ring outlined above. The stable cohomotopy is in a sense the natural analog to complex K-theory, which is denoted ${\displaystyle KU^{*}}$. Segal was inspired to make his conjecture after Michael Atiyah proved the existence of an isomorphism

${\displaystyle KU^{0}(BG)\to {\widehat {R}}[G]}$

which is a special case of the Atiyah–Segal completion theorem.

• Adams, J. Frank (1980). "Graeme Segal's Burnside ring conjecture". Topology Symposium, Siegen 1979. Lecture Notes in Mathematics. Vol. 788. Berlin: Springer. pp. 378–395. MR 0585670.
• Carlsson, Gunnar (1984). "Equivariant stable homotopy and Segal's Burnside ring conjecture". Annals of Mathematics. 120 (2): 189–224. doi:10.2307/2006940. JSTOR 2006940. MR 0763905.