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 differences and, a bit later, within the appealing mounted aspect theorem ofP. A. Smith for high periodic maps on homology spheres. Upon evaluating the fastened aspect theorem of Smith with its predecessors, the mounted aspect theorems of Brouwer and Lefschetz, one unearths that it really is attainable, not less than for the case of homology spheres, to improve the belief of mere life (or non-existence) to the particular selection of the homology form of the fastened element set, if the map is thought to be best periodic. The pioneer results of P. A. Smith basically indicates a fruitful normal path of learning topological transformation teams within the framework of algebraic topology. obviously, the instant difficulties following the Smith fastened aspect theorem are to generalize it either towards exchanging the homology spheres by means of areas of extra normal topological varieties and towards exchanging the gang tl by way of extra basic compact groups.

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I) In the case G = 7L;, the Serre spectral sequence may have non-zero differentials. However, if one assumes E2 = Eoo. then the above result holds except for the evenness of degree. We refer to a paper of 1. C. Su [S 13J for a thorough discussion of this case. (ii) In a recent paper of Tomter [T 1J, he constructs examples realizing all the above four possibilities. 1). S) [S4J. Let G= T' and X be a G-space with vanishing evendimensional rational homotopy groups. Then the fixed point set F (if non-empty) is always connected.

Pn are respectively the elementary symmetric polynomials in xJ. Example 2'. G=SO(2n). Then Wacts on r by permuting 8's and changing even number of signs. Hence e=x I ·x 2···Xn is also fixed under Wand H*(BSO (2n);

1 . Sill. T" = cosO. 0 1 1 -------------~---, I 1 L _ _ _ ~ ..... I------- ____ , 1 cosO,-sinO, 1. ~~ __ J, 1 is a maximal torus and Wacts on r by permuting 8's and changing signs. Hence xI, is the ring of symmetric polynomials in where the universal Pontrjgin classes PI,P2' ... , Pn are respectively the elementary symmetric polynomials in xJ. Example 2'. G=SO(2n). Then Wacts on r by permuting 8's and changing even number of signs. Hence e=x I ·x 2···Xn is also fixed under Wand H*(BSO (2n);