By Wu Yi Hsiang (auth.)

Historically, purposes of algebraic topology to the learn of topological transformation teams have been originated within the paintings of L. E. 1. Brouwer on periodic changes and, a bit later, within the appealing fastened aspect theorem ofP. A. Smith for top periodic maps on homology spheres. Upon evaluating the fastened element theorem of Smith with its predecessors, the fastened aspect theorems of Brouwer and Lefschetz, one unearths that it's attainable, at the least for the case of homology spheres, to improve the belief of mere lifestyles (or non-existence) to the particular choice of the homology kind of the fastened element set, if the map is thought to be top periodic. The pioneer results of P. A. Smith in actual fact indicates a fruitful basic path of learning topological transformation teams within the framework of algebraic topology. clearly, the fast difficulties following the Smith mounted aspect theorem are to generalize it either towards changing the homology spheres by means of areas of extra basic topological forms and towards changing the crowd tl by way of extra common compact groups.

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**Example text**

Now, let us consider the case R=H*(BG)=H~(Pt) and M =H2;(X, Y). 1) (Localization). Let G be a compact Lie group and X be a compact G-space. Then the localized restriction homomorphism is an isomorphism. Proof (i) We shall first prove the special case X S= 0. In this case, we need only to show that there exists SES such that n1'(s)=O in H2;(X). By the slice theorem [cf. Th. (I. 5), § 3, Ch. I], each orbit G(x) has invariant open neighborhood V such that G(x) is an equivariant retract of V. By compactness of X, there are finite number of such neighbornhoods {V 1' ...

1 E H~(X, k) is "torsion-free". Proof Suppose F =f. f/J and q E F. Then BG - {q} G ~ X G a cross-section of the fibration X G - BG and consequently IS obviously H~(Pt,k)=H*(BG,k)-H~(X,k) must be a monomorphism. On the other hand, if the above map is a monomorphism, then 1 E H~(X, k) is torsion free and hence S-I H~(X, k) =f. O. f/J. 0 46 Chapter IV. Orbit Structure and Ideal Invariants of H~(X) Corollary 2. dimkH*(F,k)~dimkH*(X,k) and equality holds if and only if' E2 =Ex for the Serre spectral sequence.

N is called the system of simple roots (w. r. t. Co). 6). (i) Let IX, [3 E LI and p, q ~ 0 be the respective largest integers such that ([3+plX), ([3-qlX)ELI. Then ([3+jlX)ELI for -q~j~p and 2([3,IJ()/(IX,IJ() =(q-p). 25 § 2. Classification of Compact Lie Groups (ii) Let 7rc,1+ be the system of simple roots. Then a;=I=a j E7r implies (a;,a)::::;O and 7r forms a basis of 9* such that every roots f3 E ,1 is an integral linear combination of simple roots with uniform sign, i. , (iii) Let 91,92 be two simple compact Lie algebra ,11' ,12 and 7r l' 7r2 be respectively their root systems and systems of simple roots.