Realization of closed manifolds as A 5 -fixed point sets

Transcription

1 Realization of closed manifolds as A 5 -fixed point sets Masaharu Morimoto Okayama University November, 2015

2 Our Problem G finite group M manifold F G (M) the family of all manifolds F obtainable as F = M G M family of smooth manifolds F G (M) = M M F G (M) 1

3 F G ({D n }) for G / P B. Oliver F G ({S n }) for various G with n G = 1 M. M. & K. Pawa lowski F G ({PC n}) various G with n G = 1 (G = A n with n 7, A 5 ) M. Kaluba In the cases above, π 1 (X) = 1 Prob. How about F G ({P n R }), F G({L 2k 1 })? For these cases, π 1 (X) < 2

4 Basic Idea due to Kaluba G nontrivial perfect group Z compact G-manifold with z 0 Z G F F({D n }) (F ) Find G-action S n with n = dim Z, S G = F F, a 0 F = diff F, T a0 S = G T z0 Z Take conn. sum Y = S n #Z at (a 0, z 0 ) 3

5 F F S n Z conn sum Z G Fig. Y = S n #Z Remove F #Z G from Y by G-surgery Question. Does there exist a G-map f : X Y s.t. f is homotopic to diffeo. & X G = F? If Answer is Yes then F F(Z) 4

6 Families of finite groups. Known Results def G G R G possesses a subquotient K/H isomorphic to a dihedral group of order 2pq for some distinct primes p and q, where H K G. G GC σ def G contains an element g being conjugate to its inverse of order pq for some distinct primes p and q. def G G C G contains an element g of order pq for some distinct primes p and q. G E def A Sylow 2-subgroup of G is not normal in G, and any element of G is of prime power order. 5

7 Thm. (B. Oliver) Let G G C E. Then F F G ({D n }) χ(f ) 1 mod n G and Case G G R : no restrictions on T F. Case G G σ C G R : c R ([T F ]) c H ( KSp(F ) ) +Tor ( KU(F ) ). Case G G C G σ C : [T F ] r C ( KU(F ) ) + Tor ( KO(F ) ). Case G E : [T F ] Tor ( KO(F ) ). G nontrivial nonsolvable = n G = 1 G E and F F G ({D n }) = dim F i = dim F j (conn. copmp. s) 6

8 Thm. (M. M. K. Pawa lowski) G nontrivial perfect group. F F G ({D n }) = F F G ({S n }) F D n double S n F F surgery F removed S n 7

9 Thm. (M. Kaluba) G nontrivial perfect group G C F F G ({D n }) & dim F i is even for some F i F if G G C G R = F F G ({P n C }) Idea. Assumption Oliver = F {a 0 } F G ({D n }) M. P. = F {a 0 } F G ({S n }) = F F G ({P n C }) S G P G C F S 2n P n C conn sum 8

10 G = A 5, F F({D n }), F i, F j F = dim F i = dim F j Thm. (M. Kaluba) G = A 5, F F G ({D n }), dim F 0 mod 2 = F F G ({P n C }) Idea. Assumption = F F F({S n }) F F S n P n C conn sum P G C P G C = P k C, k = dim F Remove F #P G C from Sn #P n C 9

11 Our Result Thm. K = R or C, W K[G]-module {X n } sequence of compact conn. manifolds s.t. G-actions on X n satisfying X G n = {x n } (one fixed pt action) T xn (X n ) = K[G] W (K[G] K) m n with lim m n n = G nontrivial perfect group G K, F F({D n }) = F F({X n }) 10

12 Flow of Modifications of G-manifolds F F({D n }), Y G-mfd s.t. Y G F Goal. Obtain X G-mfd s.t. X G = F and X = diff Y (1) Costruct f : X Y G-framed map s.t. X G = F (2) Modify f so that f H : X H Y H is diffeo (E H < G) (3) Replace f by f : X Y (by G-surgery on free part) s.t. X = diff Y 11

13 Subgroup Category and Inverse Limit G finite group S = S(G) = {H H G} S the category: Obj(S) = S Mor(S) = H,K G Mor(H, K) Mor(H, K) = {(H, g, K) g G, ghg 1 K} (K, g, L) (H, g, K) = (H, g g, L) A the category of abelian groups M : S A contravariant functor 12

14 For F = (A, B) s.t. A S, B Mor(S) Inverse limit M F = lim M( ) is F M F = { (x H ) H M(H) f1 x K = f2 x L H A for f 1, f 2 B, K, L A, f 1 = (H, g 1, K), f 2 = (H, g 2, L) B } K g 1 H g 5 2 L For lower-closed, conj.-invariant F S F = F(G, F) full subcategory of S s.t. Obj(F) = F 13

15 G-Framed Maps and G-Surgery G-framed map f = (f, b) consists of f : (X, X) (Y, Y ) b : T X ε X (R m ) f T Y ε X (R m ) 14

16 Subgroup System of A 5 H 1 = A 4, H 2 = (1, 3, 4, 2, 5), (1, 2)(3, 4) = D 10, H 3 = (1, 2, 5), (1, 2)(3, 4) = D 6, H 4 = (1, 3, 4, 2, 5) = C 5, H 5 = (1, 2)(3, 4), (1, 3)(2, 4) = D 4, H 6 = (1, 2, 5) = C 3, H 6 = (1, 2, 4) = C 3, H 7 = (1, 2)(3, 4) = C 2, H 8 = E = {e} a = (1, 2)(4, 5) 15

17 A 5 H 1 H5 H2 H 3 conj. H 6 H 6 H 7 H 4 E q 1 = (H 2, H 7, e, H 1 ), q 2 = (H 3, H 7, e, H 1 ), q 3 = (H 3, H 6, a, H 1 ) M = {H 1, H 2, H 3 } 16

18 G = A 5 M : S A contravariant functor s.t. (1) (H, g, H) = id (H G, g H) (2) (H, g, H) = id (H G, g C G (H)) M = ( M, 3 i=1 {(H, e, L), (H, g, K) (L, H, g, K) = q i } ) M M = {(x T ) T M x T M(T ), (H.e.L) x L = (H, g, K) x K, for (L, H, g, K) = q i, i = 1, 2, 3} Lem. For F = S {G}, F = F(G, F), M F = M M 17

19 Equiv Cohomotopy and Equiv Surgery Let ωg n (Y ) denote equiv stable cohomotopy group Sketch. 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 (Y ) F H F ω n H (Y ) F = S {G}, F = F(G, F) ω 0 G (pt) = A(G) the Burnside ring Q n G,F (Y ) is important to obtain G-framed maps 18

20 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 ) Def. ω 0 G def = ω 0 G (pt) 19

21 Def. A(G) = {[X] X finite G-CW complex} [X 1 ] = [X 2 ] def χ(x H 1 ) = χ(xh 2 ) ( H G) N(G, Y ) = {f = (f, b) f : X Y } where f is G-framed map Thm. Φ : ω 0 G (Y ) = N(G, Y )/ G-cob 20

22 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 ) Def. x F y def {h H : α H β H F} s.t. h H H (H, g, K) # h K rel. α, β where (H, g, K): H, K F, ghg 1 K Def. Q n G,F (Y ) = ωn G (Y )/ F, Q n G,F = ωn G / F ω n G (Y ) q F Q n G,F (Y ) res F res F Im(res F ) ω n (Y ) F lim F ωn (Y ) 21

23 For H G, χ H : A(G) Z; χ H (x) = X H 1 XH 2 (x = [X 1] [X 2 ] A(G)) A(G) def = Im ( H χ H : A(G) H G Z ) Lem. (Burnside Congrulence) (x H ) ( H G Z ) G lies in A(G) s W H x K 0 mod W H ( H G) where W H = N G (H)/H, s = K/H with H K N G (H) 22

24 (*) G nontrivial perfect group, F = S(G) {G}, F = F(G, F) Lem. (*) = β G A(G) = ω 0 G s.t. χ G (β G ) = 1 and χ K (β G ) = 0 for all K < G. Observ. finite G-CW complex D s.t. D G = & χ(d H ) = 1 (H < G) Then β G = [G/G] [D] 23

25 (*) G nontrivial perfect group, F = S(G) {G}, F = F(G, F) Thm. (Y. Hara M. M.) (*) = q F (β G ) = 0 in Q 0 G,F Cor. (*) = q F (β G ωg n (Y )) = {0} ( Qn G,F (Y )) Thm. (Y. Hara M. M.) (*) = Q 0 G,F ω0 F is injective 24