---
tag: DE29ZC399Z
kind: statement
role: theorem
name: Segal conjecture
msc: [55P91, 55Q10, 19A22]
uses: [7XRBR7AV3C, HGZAYPQBXW]
about: [7XRBR7AV3C]
sources:
  - url: https://en.wikipedia.org/wiki/Segal's_conjecture
same_as:
  wikidata: Q7446189
  wikipedia: Segal's conjecture
---

## Statement

Let $G$ be a finite group, let $BG$ be its [[classifying space|7XRBR7AV3C]], and let $BG_+$ be $BG$ with a disjoint basepoint added.
Let $A(G)$ be the Burnside ring of $G$.
Let $I \subseteq A(G)$ be its augmentation ideal, the kernel of the ring map $A(G) \to \Z$ that sends a finite $G$-set to its size.
Let $\pi^0_s$ denote stable cohomotopy in degree $0$.
Then there is a natural isomorphism
$$\pi^0_s(BG_+) \cong A(G)^\wedge_I,$$
where $A(G)^\wedge_I = \varprojlim_k A(G)/I^k$ is the $I$-adic completion.

## Remarks

The map comes from the natural ring map $A(G) \to \pi^0_s(BG_+)$, which sends a finite $G$-set $S$ to the transfer of the covering $EG \times_G S \to BG$.
The theorem says that this map induces an isomorphism after $I$-adic completion.
Lin proved the case $G = \Z/2$ and Gunawardena the case $G = \Z/p$ for $p$ odd.
Carlsson proved the conjecture for all finite groups in 1984, after a reduction to elementary abelian $p$-groups.
The analogous statement for complex representation rings and K-theory, $K^0(BG) \cong R(G)^\wedge_I$, is the Atiyah–Segal completion theorem, which motivated the conjecture.

At work: for the trivial group, $A(G) = \Z$ and $I = 0$, and the statement is $\pi^0_s(S^0) \cong \Z$, the degree of stable self-maps of spheres.

## Explanations

### L1 human

The Burnside ring of a finite group is built from the finite sets on which the group acts.
Stable cohomotopy is the most basic generalized cohomology theory, built from maps to spheres.
The Segal conjecture says that the stable cohomotopy of the classifying space of $G$ is the Burnside ring, after a completion.
So pure algebra about $G$-sets computes a hard homotopy-theoretic invariant.

### L2 human

The theorem has a stronger equivariant form: the map from the fixed points of the $G$-equivariant sphere spectrum to its homotopy fixed points becomes an equivalence after completion at $I$.
It gives a description of the stable maps between classifying spaces of finite groups, and it is a model for later completion and descent theorems in equivariant homotopy theory.
It is parallel to the [[Sullivan conjecture|9EA7KJ50AE]], which compares fixed points and homotopy fixed points for spaces instead of spectra.