A note on the Borel formula

Dotzel, Ronald M.

doi:10.1090/S0002-9939-1980-0556637-X

A note on the Borel formula
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by Ronald M. Dotzel

Proc. Amer. Math. Soc. 78 (1980), 585-589

DOI: https://doi.org/10.1090/S0002-9939-1980-0556637-X

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Abstract:

A new proof of the Borel formula is obtained for $G = {({Z_p})^r}$ actions on spaces X having ${H_i}(X;{Z_p}) = 0,i \ne n$ (some n) and ${H_n}(X;{Z_p}) = {Z_p} \oplus$ Free ${Z_p}G$ module. Each ${X^H}$ must be a ${Z_p}$-homology $n(H)$-sphere and then $n - n(G) = \Sigma (n(H) - (G))$, sum running over corank 1 subgroups. A discussion of examples follows.

References

James E. Arnold Jr., A solution of a problem of Steenrod for cyclic groups of prime order, Proc. Amer. Math. Soc. 62 (1976), no. 1, 177–182 (1977). MR 431150, DOI 10.1090/S0002-9939-1977-0431150-9
Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Ronald M. Dotzel, A converse of the Borel formula, Trans. Amer. Math. Soc. 250 (1979), 275–287. MR 530056, DOI 10.1090/S0002-9947-1979-0530056-0
Lowell Jones, The converse to the fixed point theorem of P. A. Smith. I, Ann. of Math. (2) 94 (1971), 52–68. MR 295361, DOI 10.2307/1970734
Robert Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR 375361, DOI 10.1007/BF02565743
Dock Sang Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700–712. MR 104721, DOI 10.2307/1970033