Vol. 15, No. 3, 1965

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Inflation and deflation for all dimensions

Ernst Snapper

Vol. 15 (1965), No. 3, 1061–1081

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 f1(y) has the same number h of elements for all y Y . We show that the cohomology groups Hr(X;G,A) and Hr(Y ;G,A) of the permutation representations (G,X) and (G,Y ) with values in a G-module A are interrelated by homomorphisms inflationr: Hr(Y ;G,A) Hr(X;G,A) and deflationr: Hr(Y ;G,A) Hr(Y ;G,A), for all r Z. The main properties of infr (inflationr) and defr (deflationr) are:

I. For all r Z, defrinfr: Hr(Y ;G,A) Hr(Y ;G,A) consists of multiplying the elements of Hr(Y ;G,A) by hq, where q 1 and q depends on r.

II. If for some r Z, Hr(Y ;G,A) is uniquely divisible by h, inf r is a monomorphism and defr is an epimorphism and Hr(X;G,A) = im(inf r) ker(def r), where denotes the direct sum of abelian groups.

III. Hr(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.

Mathematical Subject Classification
Primary: 20.50
Secondary: 18.00
Received: 5 March 1964
Published: 1 September 1965
Ernst Snapper