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Segal's Burnside Ring Conjecture

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Segal's Burnside Ring Conjecture Segal's Burnside Ring Conjecture Outline: I Atiyah's Theorem I Classifying spaces I Weak form of Segal's Conjecture I Strong form of Segal's Conjecture I Relation to algebraic K-theory I The Segal conjecture for tori I Towards motivic Segal conjectures. Topology Seminar, Bergen, January 24, 2013 Morten Brun, Universitetet i Bergen Atiyah's Theorem G finite group BG its classifying space KU∗ complex periodic K-theory R(G) complex representation ring of G I kernel of augmentation ": R(G) ! Z given by dimension. Theorem (Atiyah, 1961) There exists an isomorphism ^ 0 R(G)I ! KU (BG) 0 KU (X ) = [X ; BU × Z], where U = [nU(n) infinite unitary group and BU its classifying space. The Representation Ring Consider the set of isomorphism classes of finite dimensional complex representations of G. Direct sum of representations makes this set into an abelian monoid. The underlying additive group of R(G) is the group-completion of this abelian monoid. Multiplication in R(G) is induced by the tensor product over C. The representation ring of the trivial group is isomorphic to Z. Forgetting the action of the group we obtain the augmentation ∼ ": R(G) ! R(e) = Z Fiber Bundles A fiber bundle consists of a surjective continous map π : E ! X together with a topological space F . The map π is required to satisfy the following local triviality condition: for every point in X , there is an open neighborhood U of that point and a homeomorphism of the form ': U × F ! π−1(U) such that for each x 2 U, (π ◦ ')(x; v) = x for every v in F The Hopf Fibration 1 1 2 Recall that RP is obtained from the 1-sphere S ⊆ R via the 0 identification x ∼ λx for λ 2 S = f1; −1g ⊆ R. Thus there is a map 1 1 ∼ 1 S ! RP = S This is a fiber bundle with fiber S0. 1 3 2 Similarly CP is obtained from the 3-sphere S ⊆ C via the 1 identification z ∼ λz for λ 2 S ⊆ C. Thus there is a map 3 1 ∼ 2 S ! CP = S This is a fiber bundle with fiber S1. Working with quaternions and octaons we obtain fiber bundles S7 ! S4 and S15 ! S8 with fibres S3 and S7 respectively. The Hopf fibration can be visualized using a stereographic projection of S3 to R3 and then compressing R3 to a ball. This image shows points on S2 and their corresponding fibers with the same color. Vector Bundles k A complex vector bundle consists fiber bundle π : E ! X with C as fiber together with a complex vector space structure on π−1(x) for each x in X . The structure maps k −1 ': U × C ! π (U) are required to satisfy the following compatibility condition: ∼ k = −1 the map v 7! '(x; v) is an isomorphism C −! π (x) of vector spaces. Principal Bundles Let G be a topological group. A principal G-bundle is a fiber bundle π : E ! X with fiber G together with an action of G on π−1(x) for each x in X such that the resulting action of G on E is continous. The structure maps ': U × G ! π−1(U) are required to satisfy the following compatibility condition: ∼ the map g 7! '(x; g) is an isomorphism G −!= π−1(x) of G-spaces. Example: The Hopf fibrations are principal fiber bundles. Classifying spaces Let G be a topological group. There exists a principal G-bundle EG ! BG such that for a nice space X (e.g. a paracompact manifold) there is a bijection fiso classes of principal G-bundles over X g =∼ [X ; BG] 1 Example: RP is the classifying space B(Z=2Z) Complex K-theory There is a bijection fiso classes of k-dim complex vector-bundles over X g =∼ fiso classes of principal GL(k; C)-bundles over X g 0 The set KU (X ) = [X ; Z × BU] is isomorphic to the group-completion of the set of iso classes of complex vector bundles over X with respect direct sum. Application: Only the spheres S1, S3 and S7 admit an H-structure. Generalization: K-theory of operator algebras. Explicit Atiyah Isomorphism A complex representation V is a homomorphism ∼ ρ: G ! Aut(V ) = GL(k; C) The inclusion U(k) ⊆ GL(k; C) gives a homotopy equivalence ' B(U(k)) −! B(GL(k; C)): Sending ρ: G ! GL(k; C) to the composite map BG ! B(GL(k; C)) ' B(U(k)) ! fkg × BU ⊆ Z × BU we obtain a homomorphism R(G) ! [BG; Z × BU] Explicit Atiyah Isomorphism Recall that I is the kernel of augmentation ": R(G) ! Z. The homomorphism 0 R(G) ! [BG; Z × BU] = KU (BG) induces the isomorphism ^ 0 R(G)I ! KU (BG) in Atiyah's Theorem. Weak form of Segal's Conjecture The classifying space BG of a finite group G is a CW -complex. This means that it is obtained by \glueing" balls of various dimensions together. BG has a subspace BG (k) obtained by glueing the balls of dimension ≤ k. Conjecture (Segal early 1970's, proved 1984 by Carlsson) There exists an isomorphism ^ =∼ 0 (k) A(G)I −! lim πS (BG+ ) k ^ Here A(G) is the Burnside ring of G, A(G)I is its completion at 0 the augmentation ideal and πS denotes stable cohomotopy. The Burnside Ring Given a finite group G, the set of isomorphism classes of finite G-sets form a commutative monoid under disjoint union. The underlying additive group of the Burnside ring A(G) is the group-completion of this commutative monoid. Multiplication in the Burnside ring is induced by the Cartesian product. Forgetting the action of G we obtain the augmentation homomorphism ∼ ": A(G) ! A(e) = Z: Smash Product Given spaces X and Y with base-points x0 2 X and y0 2 Y we define the based sum X _ Y := X × fy0g [ fx0g × Y ⊆ X × Y and the smash-product X ^ Y := X × Y =X _ Y : The Spanier-Whitehead Category ∼ Let R[G] = map(G; R) be the regular representation of the finite group G. G We write S for the one-point compactification of R[G], and we write SnG for the iterated smash product of n factors of SG . Given finite pointed G-CW-complexes X and Y we define nG nG fX ; Y gG = colimn[X ^ S ; Y ^ S ]G Here [X ; Y ]G is the set of G-homotopy classes of maps from X to Y . The set fX ; Y gG is the hom-set in the Spanier-Whitehead category for G. Stable homotopy We have defined nG nG fX ; Y gG = colimn[X ^ S ; Y ^ S ]G Letting G act trivially on Sn, we define G n n n πn (X ) = fS ; X gG and πG (X ) = fX ; S gG Strong form of Segal's Conjecture G 0 ∼ It is a result of tom Dieck that π0 (S ) = A(G) as rings. The abelian group fX ; Y gG is a module over A(G). ∼ If G acts freely on K, then fK+; Y gG = fK=G+; Y g. BG = EG=G for a G-free G-CW complex EG, so there is an (k) ∼ (k) isomorphism fEG+ ; Y gG = fBG+ ; Y g. The weak form of Segal's Conjecture is equilvalent to Conjecture (Strong form of Segal's Conjecture) ∗ 0 ∗ (k) The map πG (S ) ! limk πG (EG+ ) induced by the map 0 EG+ ! S becomes an isomorphism after I -adic completion. When G is a p-group, the Segal Conjecture can be formulated in terms of p-adic completion. Spectra For the purpose of relating the Segal Conjecture to algebraic K-theory I use the notion of spectra. A G-spectrum consists of pointed G-spaces X0; X1;::: and G structure maps σk : S ^ Xk ! Xk+1. Given a subgroup H of G and an integer n we define the stable homotopy group H n kG πn (X ) = colimk [S ^ S ; Xk ]H : Example nG The sphere spectrum S has Sn = S . Fixed Point Spectra Given pointed G-spaces K and L, there is a G-space LK of pointed maps f : K ! L with (gf )(k) = g(f (g −1k)). Given a spectrum X and a pointed G-space K we obtain a K K K spectrum X with (X )n = (Xn) . For a p-group G the Segal Conjecture equivalent to the following: Theorem S0 EG The map S = S ! S + induces an isomorphism on p-adically ∼ G ^ = G EG+ ^ completed stable homotopy groups: πn (S)p −! πn (S )p G hG This result can be phrased as a map S ! S from the fixed point spectrum of S to the homotopy fixed point spectrum of S being a p-adic stable equivalence. Relation to algebraic K-theory There are several different categories of spectra sharing the same homotopy category. Some of these have a symmetric monoidal smash-product. Working in a sophisticated category of spectra, we can form the Hochschild complex of a ring-spectrum R: THH(R) = jR ⇔ R ^ R R ^ R ^ R · · · j This is a cyclic object in the sense of Connes, so it has an action of the circle group T. The fixed points THH(R)Cpn of THH(R) under the cyclic p-group Cpn are related to the algebraic K-theory of R. However the homotopy fixed points THH(R)hCpn are easier to analyze. The Segal Conjecture for THH Question Is the map THH(R)Cpn ! THH(R)hCpn a p-adic equivalence? For R = S, this is Segal's conjecture.
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