Younggi Choi and Seonhee Yoon

Transcription

1 J. Korean Math. Soc. 39 (2002), No. 1, TORSION IN THE HOMOLOGY OF THE DOUBLE LOOP SPACES OF COMPACT SIMPLE LIE GROUPS Younggi Choi and Seonhee Yoon Abstract. We study the torsions in the integral homology of the double loo sace of the comact simle Lie grous by determining the higher Bockstein actions on the homology of those saces through the Bockstein lemma and comuting the Bockstein sectral sequence. 1. Introduction It is well-known from the Morse theory that the integral homology of the single loo sace of any comact simle Lie grou is concentrated on even dimensions and of torsion free [1]. However the situation is different when we consider the double loo saces of comact simle Lie grous. Here we study -torsions in the integral homology of the double loo sace of the comact simle Lie grous. The main tool of the comutation is the Bockstein lemma by which we determine the higher Bockstein actions on the homology of those saces. The Bockstein sectral sequence is alied to the study of torsions in the integral homology of those saces. In this aer we get following results. For classical comact Lie grous, G = SO(n), SU(n), S(n), the order of -torsions in Ω 2 G deends on n for an odd rime. If = 2, the order of 2-torsions in Ω 2 SU(n) deends on n and the order of 2-torsions in Ω 2 S(n) and Ω 2 SO(n) is bounded by 2 or 2 2. Results are described in Theorem For the excetional Lie grous G, the order of -torsions in Ω 2 G is at most 2 3 if = 2, and at most 2 if is an odd rime. Received November 2, Mathematics Subject Classification: 55S20, 55T05, 57T10. Key words and hrases: torsion, Bockstein sectral sequence, Lie grou, loo sace. The first author was suorted by Korea Research Foundation Grant(KRF D00024).

2 150 Younggi Choi and Seonhee Yoon These are summarized in the following two tables. H (Ω 2 X; Z) condition order of 2 torsion H (Ω 2 SO(8n + i); Z) i 2, 6 2 H (Ω 2 SO(8n + i); Z) i = 2, 6 2 or 2 2 H (Ω 2 SU(n); Z) 2 r 1 < n 2 r 2 r H (Ω 2 S(n); Z) 2 H (Ω 2 G 2 ; Z) 2 H (Ω 2 F 4 ; Z) 2 H (Ω 2 E 6 ; Z) 2 or 2 3 H (Ω 2 E 7 ; Z) 2 H (Ω 2 E 8 ; Z) 2 Table 1. 2 torsion H (Ω 2 X; Z) condition order of torsion H (Ω 2 SO(2n); Z) r 1 < 2n 1 r r H (Ω 2 SO(2n + 1); Z) r 1 < 2n + 1 r r H (Ω 2 SU(n); Z) r 1 < n r r H (Ω 2 S(n); Z) r 1 2n 1 < r r H (Ω 2 G 2 ; Z) 5 = = 3 3 H (Ω 2 F 4 ; Z) 5 11 or 2 > 11 = 3 3 H (Ω 2 E 6 ; Z) 5 11 or 2 > 11 = 3 3 or 3 2 H (Ω 2 E 7 ; Z) 5 17 or 2 > 17 < 7 H (Ω 2 E 8 ; Z) 7 29 or 2 > 29 Table 2. torsion for odd rimes

3 2. Preliminaries Torsion in the homology of the double loo saces 151 Consider the short exact sequence of coefficients grous Z which induces the exact coule Z Z/() H (X; Z) k H (X; Z) ρ H (X; F ) where ρ is induced by the reduction homomorhism and k is the Bockstein homomorhism in the long exact sequence. From this exact coule we get the homology Bockstein sectral sequence {B r, β r } [1] where β r is the r-th differential in the Bockstein sectral sequence. Then β r (x) = y imlies that there exists v H (X; Z) such that k(x) = r 1 v where y = ρ(v). Hence if β r (x) = y is nonzero differential and y = s, then we have -torsion of order r in H s (X; Z). By the universal coefficient theorem, the mod homology has a following slitting: H q (X; F ) = H q (X; Z) F Ext(H q 1 (X; Z), F ). Hence a summand Z/( n ) in H q (X; Z) gives rise to summands F in H q (X; F ) and H q+1 (X; F ). Conversely if there are summands F in H q (X; F ) and H q+1 (X; F ), we can recover a summand in H q (X; Z) by determining the higher Bockstein actions in the Bockstein sectral sequence. Now we mention the homology version of the Bockstein lemma [9, 11]. Theorem 2.1. Let (E,, B; F ) be a fiber sace. Let the element u n+1 H n+1 (B; F ) be transgressive, and suose that for some integer i(i 1) and some v n+1 H n+1 (F ; F ), u n+1 transgresses to β i v n+1. Then β i+1 j (v n+1 ) is defined, and moreover (β i+1 j (v n+1 )) = β 1 u n+1 with the indeterminacy β 1 (H n+1 (E; F )) where j is the inclusion from F into E.

4 152 Younggi Choi and Seonhee Yoon 3. Torsions in the classical Lie grous Let E(x) be the exterior algebra on x and Γ(x) the divided ower algebra on x which is free over γ i (x) as a F module with roduct γ i (x)γ j (x) = ( ) i+j j γi+j (x). We have homology oerations, Dyer Lashof oerations, Q i( 1) on the (n + 1)-loo sace Ω n+1 X Q i( 1) : H q (Ω n+1 X; F ) H q+i( 1) (Ω n+1 X; F ) for 0 i n when = 2, and for 0 i n and i + q even when > 2. They are natural with resect to (n + 1)-loo mas [6]. In articular, we have Q 0 x = x. The iterated ower Q a i denotes comosition of Q i s a times. If G is a Lie grou, G is homotoy equivalent to ΩBG. Hence Q 2( 1) is defined in H (Ω 2 G; F ). In this aer the subscrit of the element always denotes the degree of that element unless stated otherwise. First we review the following basic fact. Lemma 3.1. For any rime, the order of torsions in H (Ω 2 S 2n+1 ; Z) is. Proof. We have H (Ω 2 S 2n+1 ; F 2 ) = F 2 [Q a 1x 2n 1 : a 0]. Now we consider the Bockstein sectral sequence. Then E 1 = H (Ω 2 S 2n+1 ; F 2 ). By Nishida relation, we have βq a+1 1 x 2n 1 = (Q a 1x 2n 1 ) 2 for each a 0. Since this Bockstein sectral sequence is a sectral sequence of an Hof algebra, we have E 2 = E(x 2n 1 ). Hence there is no higher differential and E 2 = E. So the 2 torsions of H (Ω 2 S 2n+1 ; F 2 ) are all of order 2. Similarly we can show the same result for odd rimes. We recall the following facts for 2 torsions. Theorem 3.2. [3] The order of 2 torsions in H (Ω 2 Sin(4n + i); Z) is 2 if i 1, and 2 or 2 2 otherwise. Since Sin(n) is a double covering sace of SO(n), we have Ω 2 Sin(n) Ω 2 SO(n). So the 2 torsions of H (Ω 2 SO(n); Z) and H (Ω 2 Sin(n); Z) are the same. For the unitary case, we have the following.

5 Torsion in the homology of the double loo saces 153 Theorem 3.3. [11] Let r and n be such that 2 r 1 < n 2 r. Then 2 r annihilates all 2 torsions in H (Ω 2 SU(n); Z), but 2 r 1 does not. Consider the following fibration: Ω 2 S(n) Ω 2 S(n + 1) Ω 2 S 4n+3. Then from [2], the corresonding Serre sectral sequence collases at the E 2 term and we have the following. Theorem 3.4. The mod 2 homology of Ω 2 S(n + 1) is H (Ω 2 S(n + 1); F 2 ) = 0 i n H (Ω 2 S 4i+3 ; F 2 ) and 2 annihilates all 2 torsions in H (Ω 2 S(n + 1); Z). Therefore 2 annihilates all 2 torsions in H (Ω 2 G; Z) where G is the following saces: S(n), Sin(4n), Sin(4n + 1), Sin(4n + 3) Now we turn to torsions for an odd rime. Combining results from [10] and [11], we have the following theorem. Theorem 3.5. (a) For an odd rime, there are choices of generators x i and y i such that H (Ω 2 SU(n + 1); F ) is isomorhic to E(Q a ( 1) x 2i 1 : a 0, 0 i n, i 0 mod ) F [Q a 2( 1) y 2i 2 : 0 i n, i 0 mod, a i > n] and β t (Q a ( 1) x 2i 1) = Q a 2( 1) y 2i 2 where t is the smallest integer such that t i > n. (b) Let be an odd rime, and r and n be integers such that r 1 < n r. Then r annihilates all torsions in H (Ω 2 SU(n); Z), but r 1 does not. We can derive the following results from the above mod homology of the double loo sace of SU(n).

6 154 Younggi Choi and Seonhee Yoon Theorem 3.6. Let be an odd rime, and r and n be integers such that r 1 < 2n 1 r. Then r annihilates all torsions in H (Ω 2 SO(2n + 1); Z), but r 1 does not. Proof. For an odd rime, we have the Harris slitting [7] SU(2n + 1) SU(2n + 1)/SO(2n + 1) SO(2n + 1) where means homotoy equivalence localized at. Hence by looing twice, we get Ω 2 SU(2n + 1) Ω 2 (SU(2n + 1)/SO(2n + 1)) Ω 2 SO(2n + 1). Therefore the mod homology of Ω 2 SO(2n + 1) for an odd rime is one of direct summands of the mod homology of Ω 2 SU(2n + 1). Moreover, we have the corresonding mod homologies: H (SU(2n + 1); F ) = E(u 2i+1 : 1 i 2n), H (SU(2n + 1)/SO(2n + 1); F ) = E(u 4i+1 : 1 i n), H (SO(2n + 1); F ) = E(u 4i 1 : 1 i n). From the above Harris slitting, we can searate H (Ω 2 SU(2n+1)) into two arts, so that H (Ω 2 SO(2n + 1); F ) is isomorhic to E(Q a ( 1) x 2i 1 : i odd, 0 i 2n 1, i 0 mod, a 0) F [Q a 2( 1) y 2i 2 : i odd, 0 i 2n 1, i 0 mod, a i > 2n 1] and β t (Q a ( 1) x 2i 1) = Q a 2( 1) y 2i 2 where t is the smallest integer such that t i > 2n 1. Therefore in the Bockstein sectral sequence, we have torsion of order t in H 2a i 2(Ω 2 SO(2n + 1); Z). Then t becomes the largest number when i = 1. Hence if r is an integer such that r 1 2n 1 < r, then r annihilates all torsions in H (Ω 2 SO(2n + 1); Z), but r 1 does not. Theorem 3.7. Let be an odd rime, and r and n be integers such that r 1 < 2n 1 r. Then r annihilates all torsions in H (Ω 2 SO(2n + 2); Z), but r 1 does not.

7 Proof. We have Torsion in the homology of the double loo saces 155 H (SO(2n + 1); Q) = E(u 4i+3 : 0 i n 1), H (SO(2n + 2); Q) = E(u 4i+1 : 0 i n 1) E(u 2n+1 ). Hence H (Ω 2 SO(2n + 2); Q) = E(u 4i 1 : 0 i n 1) E(u 2n 1 ). Consider the Serre sectral sequence converging to H (Ω 2 SO(2n); F ) Ω 2 SO(2n + 1) i Ω 2 SO(2n + 2) Ω 2 S 2n+1. From the knowledge of the rational homology, H (Ω 2 SO(2n+2); Q), we can derive that the first differential from x 2i 1 is trivial where H (Ω 2 S 2n+1 ; F ) = E(Q a 1x 2i 1 : a 0) F [βq a 1x 2i 1 : a > 0]. From commutativity between transgressions and homology oerations, the Serre sectral sequence converging to H (Ω 2 SO(2n+2); F ) collases at the E 2 term. Hence we have H (Ω 2 SO(2n + 2); F ) = H (Ω 2 SO(2n + 1); F ) H (Ω 2 S 2n+1 ; F ) and the conclusion follows. For an odd rime, we have the following equivalence SO(2n + 1) S(n) which was conjectured by Serre and roved by Harris [7]. Hence we have the following. Theorem 3.8. Let be an odd rime, and r and n be integers such that r 1 < 2n 1 r. Then r annihilates all torsions in H (Ω 2 S(n); Z), but r 1 does not.

8 156 Younggi Choi and Seonhee Yoon 4. Torsions in the excetional Lie grous The excetional Lie grous when localized at slit as followings [8]: G 2 = 3 B 2 (3, 11), = 5 B(3, 11), > 5 S 3 S 11, F 4 = 5 B(3, 11) B(15, 23), = 7 B(3, 15) B(11, 23), = 11 B(3, 23) S 11 S 15, > 11 S 3 S 11 S 15 S 23, E 6 = 5 F 4 B(9, 17), > 5 F 4 S 9 S 17, E 7 = 5 B(3, 11, 19, 27, 35) B(15, 23), = 7 B(3, 15, 27) B(11, 23, 35) S 19, = 11 B(3, 23) B(15, 35) S 11 S 19 S 27, = 13 B(3, 27) B(11, 35) S 15 S 19 S 23, = 17 B(3, 35) S 11 S 15 S 19 S 23 S 27, > 17 S 3 S 11 S 15 S 19 S 23 S 27 S 35, E 8 = 7 B(3, 15, 27, 39) B(23, 35, 47, 59), = 11 B(3, 23) B(15, 35) B(27, 47) B(39, 59), = 13 B(3, 27) B(15, 39) B(23, 47) B(35, 39), = 17 B(3, 35) B(15, 47) B(27, 59) S 23 S 39, = 19 B(3, 39) B(23, 59) S 15 S 27 S 35 S 47, = 23 B(3, 47) B(15, 59) S 23 S 27 S 35 S 39, = 29 B(3, 59) S 15 S 23 S 27 S 35 S 39 S 47, > 29 S 3 S 15 S 23 S 27 S 35 S 39 S 47 S 59. The sace B(2n 1 +1,..., 2n r +1) is built u from fibrations involving local sheres of the indicated dimensions and equivalent to a direct factor of the localization of SU(n + )/SU(n). The sace B(2n + 1, 2n + 2 1) is equivalent to a direct factor of the localization of SU(n + )/SU(n) and the cohomology of B(2n + 1, 2n + 2 1) is H (B(2n + 1, 2n + 2 1); F ) = E(x 2n+1, x 2n+2 1 )

9 Torsion in the homology of the double loo saces 157 with P 1 x 2n+1 = x 2n+2 1. From the above slitting, we get the following results immediately. Theorem 4.1. annihilates all torsions in H (Ω 2 G; Z) for > 5 G = G 2, > 11 G = F 4, E 6, > 17 G = E 7, > 29 G = E 8. From now on we denote H (Ω 2 S n ; F ) by Ω 2 (n) and r k=1 H (Ω 2 S n k ; F ) by Ω 2 (n 1,, n r ). The cases of H (Ω 2 G 2 ; F ) and H (Ω 2 F 4 ; F ) follow from [4] and the following homotoy equivalence, Ω 2 Sin(7) 2 Ω 2 G 2 Ω 2 S 7, Ω 2 Sin(7) Ω 2 SO(7). Theorem 4.2. The homology of Ω 2 G 2 is (a) H (Ω 2 G 2 ; F 2 ) = E(z 1 ) F 2 [βz 7 ] F 2 [Q a 1z 7 ) : a 0] Ω 2 (11). (b) H (Ω 2 G 2 ; F 3 ) = Ω 2 (3, 11). Theorem 4.3. The homology of Ω 2 F 4 is (a) H (Ω 2 F 4 ; F 2 ) = E(z 1 ) F 2 [βz 7 ] Ω 2 (9, 11, 15, 23). (b) H (Ω 2 F 4 ; F 3 ) = E(z 1 ) F 3 [βz 17 ] Ω 2 (11, 15, 19, 23). Corollary 4.4. For rimes = 2, 3, annihilates all torsions in H (Ω 2 G; Z) for G = G 2, F 4. Proof. We consider the Bockstein sectral sequence converging to H (Ω 2 G 2 ; Z)/torsion F 2 with E 1 = H (Ω 2 G 2 ; F 2 ). With the first Bockstein differentials, we have E 2 = E(z 1, z 9 ). Hence there is no higher differential, so that E 2 = E. So 2 annihilates all 2 torsions in H (Ω 2 G 2 ; Z). For odd rimes, we also have E 2 = E(z 1, z 9 ) and E 2 = E. Note that H (G 2 ; Q) = E(z 3, z 11 ). Similar roof works for G = F 4. We recall the following theorem in [5].

10 158 Younggi Choi and Seonhee Yoon Theorem 4.5. The homology of the double loo sace of E 6, E 7 and E 8, are as follows: (a) H (Ω 2 E 6 ; F 2 ) = E(z 1 ) F 2 [β 3 Q 2 1z 7 ] ( a 0 (E(Qa 1z 7 ) F 2 [Q a 2z 62 ])) Ω 2 (11, 15, 23), where β 3 Q a+3 1 z 7 = Q a 2z 62, a 0, H (Ω 2 E 6 ; F 3 ) = E(z 1 ) F 3 [z 16 ] ( a 0 (E(Qa 2z 17 ) F 3 [βq a+1 2 z 17 ])) Ω 2 (9, 11, 15, 17, 23). (b) H (Ω 2 E 7 ; F 2 ) = E(z 1 ) F 2 [βz 31 ] F 2 [Q a 1z 31 : a 0]) Ω 2 (11, 15, 19, 23, 27, 35), H (Ω 2 E 7 ; F 3 ) = E(z 1 ) ( a 0 (E(Qa 2z 17 ) F 3 [β 2 Q a+1 2 z 17 ])) Ω 2 (11, 15, 23, 27, 35). (c) H (Ω 2 E 8 ; F 2 ) = E(x 1 ) E(z 13 ) F 2 [βz 31, βz 55 ] ( a 0 F 2 [Q a 1z 31, Q a 1z 55 ]) Ω 2 (23, 27, 35, 39, 47, 59), H (Ω 2 E 8 ; F 3 ) = E(z 1 ) F 3 (βz 53 ) ( a 0 (E(Qa 2z 53 ) F 3 [βq a+1 2 z 53 ])) Ω 2 (15, 23, 27, 35, 39, 47, 59), H (Ω 2 E 8 ; F 5 ) = E(z 1 ) F 5 [βz 49 ] ( a 0 (E(Qa 4z 49 ) F 5 [βq a+1 4 z 49 ])) Ω 2 (15, 23, 27, 35, 39, 47, 59). From the Bockstein sectral sequence, we get the following corollaries. Corollary 4.6. The order of 2 torsions in H (Ω 2 E 6 ; Z) is 2 or 2 3 and the order of 3 torsions is 3. Proof. We consider the Bockstein sectral sequence with E 1 = H (Ω 2 E 6 ; F 2 ). With the nontrivial first differentials, we have E 2 = E(z 1 ) F 2 [β 3 Q 2 1z 7 ] ( a 0 (E(Qa 1z 7 ) F 2 [β 3 Q a+3 1 z 7 ])) E(z 9, z 13, z 21 ). Since there is no nontrivial second Bockstein differential, we have E 2 = E 3. With the third Bockstein differentials, we have E 4 = E(z 1, z 7, z 9, z 13, z 15, z 21 ). Hence there is no higher differential and E 4 = E. So the order of 2 torsions in H (Ω 2 E 6 ; Z) is 2 or 2 3. Next we consider the Bockstein sectral sequence with E 1 = H (Ω 2 E 6 ; F 3 ). Then after the first Bockstein differentials, we have E 2 = E(z 1, z 7, z 9, z 13, z 15, z 21 ). So 3 torsions of H (Ω 2 E 6 ; Z) are of order 3. Note that H (E 6 ; Q) = E(z 3, z 9, z 11, z 15, z 17, z 23 ).

11 Torsion in the homology of the double loo saces 159 Similarly we get the following two corollaries from the Bockstein sectral sequence. Corollary 4.7. The order of 2 torsions in H (Ω 2 E 7 ; Z) is 2 and the order of 3 torsions is 3 or 3 2. Corollary 4.8. The order of torsions in H (Ω 2 E 8 ; Z) is for = 2, 3, 5. Now the remaining cases are as follows: = 5, G = G 2, 5 11, G = F 4, E 6, 5 17, G = E 7, 7 29, G = E 8. In order to get torsions of H (Ω 2 G; Z) for the above cases, we should study torsions in H (Ω 2 B(3, 2 + 1); Z). Theorem 4.9. For an odd rime, H (Ω 2 B(3, 2 + 1); F ) = E(Q a ( 1) z 1 : a 0) F [β 2 Q a ( 1) z 1 : a 2]). Proof. First we comute H (ΩB(3, 2 + 1); F ). In the Eilenberg Moore sectral sequence converging to H (ΩB(3, 2 + 1); F ), we have E 2 = Tor H (B(3,2+1);F )(F, F ) = Γ(σx 3, σx 2+1 ). Then it collases at the E 2 term because E 2 is concentrated on even degrees. From the Steenrod oeration on x 3 and the Cartan formula, we have (γ i(σx 3 )) = γ i(σx 2+1 ). So the element γ i(σx 3 ) generates a truncated olynomial algebra of height 2 and H (ΩB(3, 2 + 1); F ) = i 0 (F [γ i(y 2 )]/(γ i(y 2 ) 2 )). Now we consider the Eilenberg Moore sectral sequence converging to H (Ω 2 B(3, 2 + 1); F ), E 2 = Ext H (ΩB(3,2+1);F )(F, F ) = E(z 2 a 1 : a 0) F [z 2 a+2 2 : a 0]

12 160 Younggi Choi and Seonhee Yoon and it collases at the E 2 -term by bidegree reason. If we consider the Serre sectral sequence corresonding to the fibration Ω 2 S 3 Ω 2 B(3, 2 + 1) Ω 2 S 2+1, we have E 2 = a 0 (E(Q a ( 1) ι 2 1) F [βq a+1 ( 1) ι 2 1]) ( a 0 (E(Q a ( 1) ι 1) F [βq a+1 ( 1) ι 1])) where H (Ω 2 S 2n+1 ; F ) = a 0 (E(Q a ( 1) ι 2n 1) F [βq a+1 ( 1) ι 2n 1]). Then there are nontrivial differentials d(q a ( 1) ι 2 1) = βq a+1 ( 1) ι 1 for a 0. Hence by the Bockstein lemma, β 2 Q a+2 ( 1) ι 1 = βq a+1 ( 1) ι 2 1. So E = E(Q a ( 1) ι 1 : a 0) F [β 2 Q a+2 ( 1) ι 1 : a 0]. Hence we have the conclusion. Corollary annihilates all torsions in H (Ω 2 B(3, 2 + 1); Z) for an odd rime. Proof. We consider the Bockstein sectral sequence for an odd rime with E 1 = H (Ω 2 B(3, 2 + 1); F ). Since there is no first Bockstein actions, we have E 1 = E 2. After the second Bockstein differentials, we have that E 3 = E(ι 1, Q ( 1) ι 1 ) and E 3 = E. So torsions of H (Ω 2 B(3, 2 + 1); F ) are all of order 2. From the above result, we can determine torsions of H (Ω 2 G; Z) for the following cases. Corollary annihilates all torsions in H (Ω 2 G; Z) for the following cases: = 5, G = G 2, 5 11, G = F 4, E 6, 5 17, G = E 7, 7 29, G = E 8.

13 Torsion in the homology of the double loo saces 161 References [1] R. Bott, The sace of loos on a Lie grou, Michigan Math. J. 5 (1958), [2] Y. Choi, The homology of Ω 2 S(n) and Ω 3 0S(n), Comm. Korean Math. Soc. 9 (1994), [3], Homology of the double and trile loo sace of SO(n), Math. Z. 222 (1996), [4] Y. Choi and S. Yoon, Homology of trile loo sace of the excetional Lie grou F 4, J. Korean Math. Soc. 35 (1998), [5], Homology of the double and the trile loo sace of the excetional Lie grou E 6, E 7, and E 8, Manuscrita Math. 103 (2000), [6] F. R. Cohen, T. Lada, and J. P. May, The homology of iterated loo saces, Lect. Notes. Math. Vol. 533, Sringer, [7] B. Harris, On the homotoy grous of the classical grous, Ann. of Math. 74 (1961), [8] M. Mimura, Homotoy theory of Lie grous, Handbook of algebraic toology edited by I. M. James, North-Holland (1995), [9] R. E. Mosher and M. C. Tangora, Cohomology oerations and alications in homotoy theory, Harer & Row, Publishers New York, Evanston, and London, [10] D. C. Ravenel, The homology and Morava K-theories of Ω 2 SU(n), Forum Math. 5 (1993), [11] D. Waggoner, Loo saces and the classical unitary grous, Ph. D. Thesis, University of Kentucky, Younggi Choi Deartment of Mathematics Education Seoul National University Seoul , Korea yochoi@laza.snu.ac.kr Seonhee Yoon Korean Information Security Agency Seoul , Korea shyoon@kisa.or.kr