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Burnside rings and fusion systems
Sune Precht Reeh
Centre for Symmetry and Deformation
Department of Mathematical Sciences
YTM 2013, July 11
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Outline
1
Burnside rings and the Segal conjecture
The Burnside ring of a finite group
The Segal conjecture
2
Fusion stable elements
The Burnside ring of a saturated fusion system
A transfer map
3
Segal conjecture for fusion systems
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The Burnside ring of a finite group
Let G be a finite group. The isomorphism classes of finite
G-sets form an abelian monoid with disjoint union as
addition.
The Grothendieck group of this monoid is the Burnside
ring of G, denoted A(G). The multiplication in A(G) is
given by Cartesian products.
Every finite G is the disjoint union of its orbits, hence the
transitive G-sets [G/H] for H ≤ G form an additive basis
for A(G).
The basis element [G/H] only depends on H up to
conjugation in G.
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For each finite G-set X and any subgroup H ≤ G, we can
count the number of fixed points |X H |. Taking fixed
points respects the structure of A(G):
|(X t Y )H | = |X H | + |Y H | and |(X × Y )H | = |X H | · |Y H |,
so the fixed point maps X 7→ |X H | extend to ring
homomorphisms A(G) → Z.
|X H | only depends on H up to G-conjugation, and the
collection of numbers |X H | for H ≤ G determines
X ∈ A(G) uniquely.
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Connections to topology
Theorem (Segal conjecture, proven by Carlsson)
Let G be a finite group. The 0’th stable cohomotopy group
πS0 (BG+ ) = [Σ∞ (BG+ ), S] is isomorphic to the completion
A(G)∧
I of the Burnside ring at a certain ideal I ≤ A(G).
This completion is in general difficult to describe.
However, if S is a p-group, then A(S)∧
I is (almost) the
p-completion A(S)∧
.
p
Question: What is the corresponding result for πS0 (BG∧
p )?
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Fusion systems
A fusion system over a finite p-group S is a category F
where the objects at the subgroups P ≤ S and the
morphisms satisfy:
• HomS (P, Q) ⊆ F(P, Q) ⊆ Inj(P, Q) for all P, Q ≤ S.
• Every ϕ ∈ F(P, Q) factors in F as an isomorphism
P → ϕP followed by an inclusion ϕP ,→ Q.
A saturated fusion system satisfies a few additional axioms
that play the role of Sylow’s theorems.
The canonical example of a saturated fusion system is
FS (G) defined for S ∈ Sylp (G) with morphisms
HomFS (G) (P, Q) := HomG (P, Q).
for P, Q ≤ S.
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The Burnside ring of a fusion system
Let F be a saturated fusion system over S. A finite S-set
X (or an element of A(S)) is said to be F-stable if
|X Q | = |X P | whenever Q, P are isomorphic/conjugate in
F.
Theorem (R.)
The isomorphism classes of F-stable finite S-sets form a
free abelian monoid. The irreducible stable sets are in
1-to-1 correspondence with the subgroups of S up to
F-conjugation.
The Grothendieck group of this monoid is the Burnside
ring of F, denoted A(F), and is a subring of A(S).
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A transfer map
Theorem (R.)
Let F be a saturated fusion system over a finite p-group S.
∧
Then there is a transfer map π : A(S)∧
p → A(F)p
satisfying
π(X)Q =
X 1
X Q0 0
#{Q ∼F Q} 0
Q ∼F Q
for all X ∈ A(S)∧
p and Q ≤ S. Furthermore, π is a
homomorphism of A(F)∧
p -modules and restricts to the
∧
identity on A(F)p .
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The characteristic idempotent
A generalization of the Segal conjecture for p-groups S, T
states that homotopy classes of stable maps [BS, BT ] is
(almost) the p-completion of A(S, T ) – which is
constructed like a Burnside ring, but from sets that have
both an S-action and a free T -action that commute.
To each saturated fusion system F there is a unique
associated element 0 6= ωF ∈ A(S, S)∧
p such that
• ωF is F-stable with respect to both S-actions,
• every stabilizer of ωF has the form {(p, ϕp) | p ∈ P }
for some F(P, S),
• the corresponding stable homotopy class
ωF ∈ [BS, BS] is idempotent.
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The Segal conjecture for fusion systems
The classifying spectrum BF can be constructed as the
infinite mapping telescope
ω
ω
F
F
BF := hocolim(BS −−→
BS −−→
BS → · · · ).
If F = FS (G) with S ∈ Sylp (G), then BF = Σ∞ (BG∧
p ).
Theorem (Ragnarsson)
We can view π 0 (BF) as a subring of π 0 (BS), and then we
have π 0 (BF) = (ωF )∗ (π 0 (BS)).
In particular, π 0 (BF) is (almost) isomorphic to
∧
∼
(ωF )∗ (A(S)∧
p ) = A(F)p .
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Theorem (R.)
∧
The map A(S)∧
p → A(F)p induced by the characteristic
∧
idempotent ωF ∈ A(S, S)p coincides with the transfer map
from earlier:
Q X 1
(ωF )∗ (X) =
X Q0 .
#{Q0 ∼F Q}
0
Q ∼F Q
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Thanks for your attention!
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References
[1] Kári Ragnarsson, Classifying spectra of saturated fusion systems,
Algebr. Geom. Topol. 6 (2006), 195–252. MR2199459
(2007f:55013)
[2] Kári Ragnarsson and Radu Stancu, Saturated fusion systems as
idempotents in the double Burnside ring, 54 pp., preprint,
available at arXiv:0911.0085.
[3] Sune Precht Reeh, The abelian monoid of fusion-stable finite sets
is free, 14 pp., preprint, available at arXiv:1302.4628.
[4] Sune Precht Reeh, Transfer and characteristic idempotents for
saturated fusion systems, 39 pp., preprint, available at
arXiv:1306.4162.
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