The Burnside ring, equivariant framed maps, and the reflection method

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1 The Burnside ring, equivariant framed maps, and the reflection method Masaharu Morimoto Okayama University July, 2015

2 This contains results obtained from discussions with Takashi Matsunaga (on V -trasversality) Yasuhiro Hara (on Burnside ring) Marek Kaluba (on Reflection Method)

3 Notation in Talk G finite group S(G) = {H H G} F S(G) s.t. (1) H F S(H) F lower closed (2) H F, g G ghg 1 F conjugation invariant F = F(G, F) category (associated with G and F) V real G-module X, Y compact G-manifolds (or finite G-CW complexes) 1

4 Sketch of Talk Let ωg n (Y ) denote equivaraint stable cohomotopy group We will introduce ω n G (Y ) q F H F res G H res F Q n G,F (Y ) res F Im(res F ) ω n F (Y ) H F ω n H (Y ) where ωf n (Y ) = inv-lim F ωh n (Y ) (H F) ω 0 G (pt) = A(G) the Burnside ring Q n G,F (Y ) is relevant to the Reflection Method 2

5 where [X] = [X ] χ H : A(G) Z; Burnside Ring A(G) A(G) = {[S 1 ] [S 2 ] S i finite G-sets} = {[X] X finite G-CW complex} def χ(x H ) = χ(x H ) for H G. χ H ([S 1 ] [S 2 ]) = S H 1 SH 2, χ H([X]) = χ(x H ) A(G) H χ H H G Z G G-conj. inv. = (H) (G) Z (H) def = {ghg 1 g G} 3

6 Cohomotopy groups ω n G (Y ) M(X, Y ) G 0 = {f : X Y b.p.pres. G-map} [X, Y ] G 0 = M(X, Y )G 0 / G-ht Y + = Y {y } V = V { } ωg n def (Y ) = {Y +, R n } G 0 = lim n [Y + V, (R n V ) ] G 0 (V = C[G]n ) (equivariant stable) cohomotopy group Y 0 y 0 Y + V (R n V ) ω 0 G def = ω 0 G (pt) 4

7 Thm. Pontryagin-Petrie Construction gives Φ : ω 0 G (Y ) 1:1 { G-framed cobordism classes [f] of G-framed maps f with f : X Y } G-framed map f is pair (f, b) of f : X Y G-map, b : T X ε(r m ) f T Y ε(r m ) G-isomorphism 5

8 Prob. Let {f H H F} be family s.t. We ask whether f H : X H Y is H-framed map. a G-framed map f : X Y s.t. res G H f H f H H F Nec. Cond. (1) res H H K f H H K res K H K f K (H, K F) (2) c # g f ghg 1 H f H (g G, H F) where c g : H ghg 1, c # g X ghg 1 = {g 1 x x X ghg 1} copy of X ghg 1 g 1 x x c # g X ghg 1 1:1 X ghg 1 c # g f ghg 1 f ghg 1 Y Y L 1 g 6

9 Inv-limit ω n F (Y ) F = F(G, F): Obj = F, Mor = {(H, g, K)}, where H, K F, g G s.t. ghg 1 K (H, g, K) : ω n K (Y ) res K ghg 1 ω n ghg 1 (Y ) c g ω n H (Y ) Def. ω n F (Y ) = inv-lim F ω n H (Y ) H F ω n H (Y ) ω n G (Y ) res F ω n F (Y ) H F res G H H F ω n H (Y ) Prob. We ask whether res F : ωg 0 (Y ) ω0 F (Y ) is surjective. 7

10 Examples Let F = S(G) {G}, F = F(G, F), ω n F def = ω n F (pt), ω 0 G = A(G), Im(res F) = Im[res F : ω 0 G ω0 F ]. E.g. G = C n (n = p m, p prime) = ω 0 F /Im(res F) = Z m 1 p. E.g. G = C p C p (p prime) = ω 0 F /Im(res F) = Z p. E.g. G = C p C q (p q primes) = ω 0 F = Im(res F). E.g. G = C p C q (p q primes) s.t. G is nonabelian = ω 0 F = Im(res F). E.g. G = A 4 = D 4 C 3 = ω 0 F = Im(res F). 8

11 Nilpotent G and Surjectivity of res F Let F = S(G) {G}, F = F(G, F) Thm. Let G be nilpotent. Then res F : ωg 0 ω0 F is surjective G = C p 1 p 2 p n for some distinct primes p 1, p 2,..., p n. Cor. Suppose G is nilpotent and Y G. If res F : ωg 0 (Y ) ω0 F (Y ) is surjective then for some distinct primes p 1, p 2,..., p n. G = C p 1 p 2 p n Proof. ωg 0 (y 0) π ωg 0 (Y ) j ωg 0 (y 0) res F res F surj ω 0 F (y 0) π ω 0 F (Y ) j res F surj ω 0 F (y 0) 9

12 Question Let F = S(G) {G}, F = F(G, F) Suppose G = p 1 p 2 p m (p i are distinct primes). Question. We wonder whether res F : ω 0 G ω0 F is surjective. 10

13 V -Transversality (due to T. Petrie) Let α : Y + V V (α(y ) = ) s.t. V R n+2, n = dim Y We say that α is V -transversal to 0 in V (α V 0) if α is smooth on X = α 1 (0) ( Y V ), dα x : T x (Y V ) T 0 (V ) is onto for x X, and ν(α) x : V H V H, H = G x, (called normal derivative) is id VH for x X, where V = V H V H. V H ν(α) x V H incl V = T x (x V ) proj T x (Y V ) dαx T 0 (V ) = V 11

14 Pontryagin-Petrie Construction Suppose α V 0 (V -transversal). Set X = α 1 (0), f : X Y V π Y Y. T (Y V ) X = (π Y T Y π V T V ) X = f T Y ε(v ) T (Y V ) X = T X ν(x, Y V ) = T X ε(v ) ( dα x : T x (Y V ) V is surjective for x X) ξ : T X ε(v ) = f T Y ε(v ) with ξ ε(vh ) = id Lem. (Lück-Madsen) b : T X ε(r n+2 ) = f T Y ε(r n+2 ). Def. f = (f, b) is called a G-framed map. Proc. α f is called the Pontryagin-Petrie Construction. 12

15 Reflection Method f : X Y, id Y : Y Y G-framed maps, H K < G F K : W K I Y K-framed cobord.: f K id Y (I = [0, 1]) Suppose F K is product on neighborhood N of W >H K (= L>H W K L ) N : W >H K N W K, N = I M, M Y, and N X = 0 M, X >H 0 M (X >H = L>H X L ) N Y = 1 M, Y >H 1 M (= M) F K N = I id M 13

16 Suppose N G (H) K Set L = N G (H) and take L-neighborhood U L = U L (W K H, W K ) Set W G = (G L U L ) (I X) 1 X F G = (G L F K UL ) W K = W G 1 X W K 0 X (I F K X ) X G L U L X I X W K Y W GW K W K H Y H F G G-framed cobord. : f G f : X Y s.t. f H = idy H 14

17 X X W W W = W X W Y = X I X W W I Y Y W W rel. X, Y s.t. W = I Y. W K : X K Y s.t. (1) W K K W K rel. X, Y, (2) W K H = I Y H. W K H may not be I Y H for H = ghg 1 ( K), g G K. X W K Y c # g W K 15

18 F K : f K id Y ω 0 G (Y ) res G K ω 0 K (Y ) res G K Φ 1 ([f]) = res G K Φ 1 ([id Y ]) where Φ 1:1 { } G-framed cobordism classes [f] of G-framed maps f : X Y resg K Φ 1 16

19 Equivalence Relation F on ω n G (Y ) x, y ω n G (Y ), x = [α], y = [β], α, β : Y + V (R n V ) x F y def {h H : α H β H F} s.t. h H H h K rel. α, β (H < K F) c # a h aha 1 H h H rel. α, β (H F, a G) x F y for x = Φ 1 ([f : X Y ]), y = Φ 1 ([id Y ]) = W K s.t. W H K = I Y H (H = ghg 1 K, g G K) Def. Q n G,F (Y ) = ωn G (Y )/ F, res F ωg n (Y ) q F Q n G,F (Y ) Im(res res F ) F Q n G,F = ωn G / F ω n F (Y ) inv-lim F ω n H (Y ) 17

20 Def. (Oliver) k G = p : p prime s.t. N G with G : N = p (If G is perfect then k G = 1.) For x = [X 1 ] [X 2 ] A(G), set χ H (x) = X1 H XH 2. Lem. (due to Oliver) γ G A(G) = ω 0 G s.t. χ G (γ G ) = k G and res G K γ G = 0 for all K < G. Prop. Let F = S(G) {G}, F = F(G, F). Then is generated by γ G. Ker[res F : ω 0 G ω0 F ] (ω0 F = inv-lim F ω 0 H ) 18

21 Let G / P, F = S(G) {G}, F = F(G, F). Thm. Ker[q F : ω 0 G Q0 G,F ] is generated by a unique element τ G of ω 0 G = A(G) s.t. χ G(τ G ) > 0. Moreover, τ G is divisible by γ G and γ m G is divisible by τ G for some m N. (γ m G = km 1 G γ G) Cor. τ G ωg n (Y ) Ker[q F : ωg n (Y ) Qn G,F (Y )] Proof. ω 0 G ωn G (Y ) ωn G (Y ) q F q F q F Q 0 G,F Qn G,F (Y ) Qn G,F (Y ) 19

22 Prop. Q 0 G,F = Im[res F : ωg 0 ω0 F ] Z u where u = χ G (τ G )/k G. Prop. If G is nontrivial perfect then k G = 1, τ G = γ G, and Q 0 G,F = Im[res F : ωg 0 ω0 F ]. Lem. Let β A(G) = ωg 0 s.t. resg Hβ = 0 ( H < G). Then m N s.t. q F (β m ) = 0. 20