We assume that a finite group
G acts on the left on finite sets X and Y , and that there is given a function
f : X → Y . We assume that f(σx) = σf(x) for all σ ∈ G and x ∈ X; and that f^{−1}(y)
has the same number h of elements for all y ∈ Y . We show that the cohomology
groups H^{r}(X;G,A) and H^{r}(Y ;G,A) of the permutation representations (G,X) and
(G,Y ) with values in a Gmodule A are interrelated by homomorphisms inflation_{r}:
H^{r}(Y ;G,A) → H^{r}(X;G,A) and deflation_{r}: H^{r}(Y ;G,A) → H^{r}(Y ;G,A),
for all r ∈ Z. The main properties of inf_{r} (inflation_{r}) and def_{r} (deflation_{r})
are:
I. For all r ∈ Z, def_{r}inf_{r}: H^{r}(Y ;G,A) → H^{r}(Y ;G,A) consists of multiplying the
elements of H^{r}(Y ;G,A) by h^{q}, where q ≧ 1 and q depends on r.
II. If for some r ∈ Z, H^{r}(Y ;G,A) is uniquely divisible by h, inf _{r} is a
monomorphism and def_{r} is an epimorphism and H^{r}(X;G,A) = im(inf _{r}) ⊕ ker(def _{r}),
where ⊕ denotes the direct sum of abelian groups.
III. H^{r}(Y ;G,A) is uniquely divisible by h for all r ∈ Z in each of the following
two cases.
IIIa. A is uniquely divisible by h.
IIIb. (h,m) = 1 where m is the index of (G,Y ).
We then study the special case where the permutation representations (G,X) and
(G,Y ) are transitive and where (G,X) is furthermore free of fixed points. Since the
classical inflation and deflation mappings fall under this heading, we have now
extended these mappings to all of Z. We describe the six mappings inf _{r} and def _{r} for
r = 0,±1 explicity in terms of trace mappings, augmentation ideals and crossed
homomorphisms.
