# Bredon cohomology

An ordinary equivariant cohomology for a finite group $G$, defined in [a1], on the category $G$- CW of $G$- complexes (cf. Complex; CW-complex). The objects of $G$- CW are the CW-complexes $X$ with a cellular action of $G$, satisfying the condition that, for every subgroup $H$ of $G$, the fixed point set $X ^ {H}$ is a subcomplex of $X$. The morphisms are the cellular $G$- mappings. Let $O _ {G}$ be the full subcategory of $G$- CW whose objects are the $G$- orbits $G/H$, where $H$ is a subgroup of $G$. For every contravariant functor $M$ from $O _ {G}$ to the category ${ \mathop{\rm Ab} }$ of Abelian groups, there exists a Bredon cohomology theory $\{ H ^ {n} _ {G} ( -,M ) \} _ {n \in \mathbf N }$ which, after restriction to $O _ {G}$, vanishes for $n > 0$ and is equal to $M$ for $n = 0$.

Let $c _ {*} ( X )$ be the chain complex of functors from $O _ {G}$ to ${ \mathop{\rm Ab} }$ such that, for every subgroup $H$ of $G$, $c _ {*} ( X ) ( G/H ) = C _ {*} ( X ^ {H} )$ is the ordinary cellular chain complex of $X ^ {H}$. Then

$$H ^ {n} _ {G} ( X,M ) = H ^ {n} ( { \mathop{\rm Hom} } _ {O _ {G} } ( c _ {*} ( X ) ,M ) ) ,$$

where ${ \mathop{\rm Hom} } _ {O _ {G} } ( -, - )$ denotes the set of natural transformations of functors. The functors $c _ {n} ( X )$ are projective objects in the category of coefficient systems and there is a spectral sequence

$${ \mathop{\rm Ext} } ^ {p} _ {O _ {G} } ( h _ {q} ( X ) ,M ) \Rightarrow H ^ {p + q } _ {G} ( X,M ) ,$$

where $h _ {q} ( X ) ( G/H ) = H _ {q} ( X ^ {H} )$.

Let $Y$ be a $G$- space with base point $y _ {0}$( cf., e.g., Equivariant cohomology). Important examples of coefficient systems are the homotopy group functors $\pi _ {n} ( Y )$ defined by $\pi _ {n} ( Y ) ( G/H ) = \pi _ {n} ( Y ^ {H} ,y _ {0} )$. The obstruction theory for $G$- mappings $f : X \rightarrow Y$ is formulated in terms of the cohomology groups $H ^ {n} _ {G} ( X, \pi _ {m} ( Y ) )$. For any coefficient system $M$ and natural number $n > 0$, there is a pointed Eilenberg–MacLane $G$- complex $K ( M,n )$ such that $\pi _ {n} ( K ( M,n ) ) = M$ and $\pi _ {m} ( K ( M,n ) )$ vanishes whenever $n \neq m$. For every $G$- complex $X$, $H ^ {n} _ {G} ( X,M ) = [ X,K ( M,n ) ] _ {G}$, where $[ -, - ] _ {G}$ denotes $G$- homotopy classes of $G$- mappings.

If $h ^ {*} _ {G} ( - )$ is an equivariant cohomology theory defined on the category $G$- CW, then there exists an Atiyah–Hirzebruch-type spectral sequence

$$H ^ {p} _ {G} ( X,h ^ {q} ) \rightarrow h ^ {p + q } _ {G} ( X ) ,$$

where $h ^ {q}$ is the restriction of $h ^ {q} _ {G} ( - )$ to $O _ {G}$. Bredon cohomology for an arbitrary topological group is studied in [a4] and [a5]. Singular ordinary equivariant cohomology is defined in [a2] (the finite case) and in [a3]. If a coefficient system $M$ is a Mackey functor, then the Bredon cohomology $H ^ {*} _ {G} ( -,M )$ can be extended to an ordinary ${ \mathop{\rm RO} } ( G )$- graded cohomology [a6].

#### References

 [a1] G.E. Bredon, "Equivariant cohomology theories" , Lecture Notes in Mathematics , 34 , Springer (1967) [a2] T. Bröcker, "Singuläre Definition der äquivarianten Bredon Homologie" Manuscr. Math. , 5 (1971) pp. 91–102 [a3] S. Illman, "Equivariant singular homology and cohomology" , Memoirs , 156 , Amer. Math. Soc. (1975) [a4] T. Matumoto, "Equivariant cohomology theories on -CW-complexes" Osaka J. Math. , 10 (1973) pp. 51–68 [a5] S.J. Wilson, "Equivariant homology theories on -complexes" Trans. Amer. Math. Soc. , 212 (1975) pp. 155–171 [a6] L.G. Lewis, J.P. May, J. McClure, "Ordinary RO(G)-graded cohomology" Bull. Amer. Math. Soc. , 4 (1981) pp. 208–212
How to Cite This Entry:
Bredon cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bredon_cohomology&oldid=46160
This article was adapted from an original article by J. SÅ‚omiÅ„ska (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article