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Algebraic Topology: A Primer (Texts and Readings in by Satya Deo

By Satya Deo

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JLL « [M; a, b; Rn-m + p, b = t Qm 4- t Sn - n + q , s' = s + ( t mR+ t nS)(q-n) + ^ m Q + SiS) (m-p) - ^ n + t qm + ^ q Proof. It follows form direct calculation. 38 MINKING EIE Let COROLLARY. Suppose M L = [M; p, q; s] be an element of has an isolated fixed point, and let characteristic polynomial; then M x. n si 0 1 , s G Z Proof. if L r« (x) be its is conjugate in Sp(3, Z) tp_

I1 [OJ s 0 0 0 , 0; , 0; , 0; ], [s 0' 0 0. [s 01 s s , 0; }] -~ S l S [s 1 2 t3 2 J or s S S s+% 0 0 0 s-% 0 12, ] ~ [M , E ; 0, 0; 3 2 l = s 12 = 12 or ], s e Z. S + 3 S s S l'' 12 12 ] - [M , E ; 0, 0; sx-% 3 2 S S 2 J 12 s l = s 12 = , s € Z. 12 ], 2 °' s 2 ^ ° S l l 2~ s12 ^ ^ [M3, E 2 ; 0, 0; ] -~ [M£, E 2 ; 0, 0; ] L° oj 1 S 2 ~ s l? ^ ° [M3, E 2; s € Z - {0} . 5 Conjugacy classes The c o n j u g a c y the following -1] R) S\ M the type (3) ,0 r~ of are s p ( 3 , R) (5) is with determined conjugate cos 8 -sin is U' = E 3 or diagtl, S = 0 "2 ' "3 ' l s 12 t S = s cos in S p ( 3 , Z) t o one of the following S e M~(Z) , U 12 s2 u 0 u s \sl u 0 u s2 V 0 v 0 S = diag[slf S = ?!

Then M Let e2ir±/9 M is conjugate in (v = e 2iri/7 ) by e2jri/7 . be an element of Sp(3, R) Sp(3, Z) or order 7, to one of the following: 24 MINKING EIE [v, v 2 , v 3 ] , [v, v 2 , v 4 ] , [v, v 4 , v 5 ] , [v, v 3 , v 5 , 5 , v6,, v ] . r 2 [v2 , v 3 , v 4, ] , [v ,4v ,6, v 3r,4 [v 5 , v 6, , v r ]3 , [v r LEMMA 6. The number of conjugacy classes of regular elliptic elements of order X -fX -2X+1 , in Sp(3, Z) a, 3, y Suppose THEOREM 3. equation 7 is are distinct roots of the (or more precisely, 4TT 8 .

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