# zbMATH — the first resource for mathematics

Smith theory and quasi-periodicity in Bredon cohomology. (English) Zbl 0762.55005
Let $$G$$ be a finite group. A $$CW$$-complex $$K$$ with a given action of $$G$$ is said to be a $$G$$-$$CW$$-complex, if for every subgroup $$H$$ of $$G$$, the fixed point set $$K^ H$$ is a subcomplex of $$K$$. In this paper, the author studies the equivariant cohomology theory of $$G$$-$$CW$$-complexes by using Bredon cohomology. Thus the author was able to generalize some well-known results in the classical Smith theory. For intance, the following special case of his results will yield Theorem 5.4 in Chapter III of G. E. Bredon’s book: Introduction to compact transformation groups (1972; Zbl 0246.57017)]. Let $$K$$ be a $$G$$-$$CW$$-complex. Assume that $$A$$ is an abelian group, and that $$q$$ and $$m$$ are natural numbers such that $$q>m$$. Suppose that, for every subgroup $$H$$ of $$G$$, $$H^ n(K/H,A)=0$$, whenever $$n=q$$, $$q-1$$, and that $$H^ n(K,A)=0$$ whenever $$m\leq n\leq q$$. Then $$H^ n(K/G,A)=0$$ whenever $$m\leq n\leq q$$.
Reviewer: H.T.Ku (Amherst)

##### MSC:
 55N91 Equivariant homology and cohomology in algebraic topology 55M35 Finite groups of transformations in algebraic topology (including Smith theory) 57S17 Finite transformation groups
Full Text:
##### References:
 [1] G.E. Bredon : Equivariant cohomology theory . Lecture Notes in Math. 34, Berlin- Heidelberg-New York: Springer 1967. · Zbl 0162.27202 [2] G.E. Bredon : Introduction to compact transformation groups . New York -London: Academic Press, 1972. · Zbl 0246.57017 [3] K.S. Brown : Cohomology of groups . New York- Heidelberg-Berlin: Springer 1982. · Zbl 0584.20036 [4] H. Cartan and S. Eilenberg : Homological Algebra . Princeton, N.J.: Princeton University Press, 1956. · Zbl 0075.24305 [5] L. Choinard : Projectivity and relative projectivity over group rings , J. Pure and Applied Algebra 7 (1976), 287-302. · Zbl 0327.20020 · doi:10.1016/0022-4049(76)90055-4 [6] P.J. Hilton : and U. Stammbach : A course in homological algebra . New York- Heidelberg-Berlin: Springer, 1970. · Zbl 0238.18006 [7] S. Jackowski : The Euler class and periodicity of groups cohomology , Comment. Math. Helvetici 53 (1978), 643-650. · Zbl 0404.20043 · doi:10.1007/BF02566105 · eudml:139762 [8] K. Jänich : Differenzierbare G-Mannigfaltgkeiten . Lecture Notes in Math. 59, Berlin- Heidelberg-New York: Springer 1968. · Zbl 0159.53701 [9] J.P. May : A generalization of Smith theory , Proc. of A.M.S. 101 (1987), 728-730. · Zbl 0635.57020 · doi:10.2307/2046679 [10] R.L. Rubinsztein : On the equivariant homotopy of spheres, Dissertationes Mathematicace 134, PWN 1976. · Zbl 0343.57021 [11] J.P. Serre : Sur la dimension cohomologique des groupes profinis , Topology 3 (1965), 413-420. · Zbl 0136.27402 · doi:10.1016/0040-9383(65)90006-6 [12] J. Słomińska : Equivariant Bredon cohomology of classifying spaces of families of subgroups , Bull Ac. Pol. Math. 28 (1980), 503-508. · Zbl 0479.55006 [13] J. Słomińska : Hecke structure on Bredon cohomology , to appear in Fundamenta Mathematicae. · Zbl 0812.55004 · matwbn.icm.edu.pl · eudml:211925 [14] J. Słomińska : Finiteness conditions in Bredon cohomology , to appear in J. Pure and Applied Algebra. · Zbl 0748.55003 · doi:10.1016/0022-4049(91)90083-E [15] J S\?Omińska : Dimension in Bredon cohomology . In preparation. · Zbl 0841.55004 · doi:10.1016/0022-4049(94)90041-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.