TITLE: What kind of geometric operations "scale up" cohomology?
QUESTION [13 upvotes]: There's an obvious operation on the category of graded rings, given by "scaling up," multiplying the grading of every element by some fixed constant.
Does anyone know of an operation on the level of spaces that will do this to the cohomology rings?

REPLY [16 votes]: How about we change the question to this: for what kind of spaces X are there a family of  continuous maps ψk: X->X which act on cohomology by "scaling up"?  Here is a famous and bewildering example of something sort of like this:
As you can read in the notes of Sullivan's 1970 MIT course (see esp. chapter 5), if you have an algebraic variety X defined over Q(the rationals), then you get an action of G(=absolute Galois group of Q) on a space Xet=the etale homotopy type of X.
If X is a Grassmanian variety, then
* Xet has the homotopy type of the usual complex Grassmannian (up to a "profinite completion"),
* G acts on the profinite cohomology of Xet through its abelianization Gab=Z*(=the units of the profinite integers),
* this action of Gab on cohomology is by scaling.
I don't know what the motivation of the original question is, so I don't know if this kind of thing has any relation for what you want.  (I really posted this to advertise the Sullivan notes, which everybody should read.)