I really very much like the averaging trick to make a continuous cocycle cohomologous to a smooth one. I believe this will give a proof of the fact the cohomology defined using continuous cochains and smooth cochains are indeed isomorphic. It is also true that in some case, there is no difference between Moore’s cohomology using Borel cochains and continuous cohomology.

Is there a simple way to see that a Borel cocycle is cohomologous to a smooth one ?

]]>So, isn’t really a cocycle in the sense of your previous blog post. But you call it cocycle so often, that I feel that is is a a real cocycle in some other sense it is. As far as I know, cocycles are cochains with zero coboundary, Is there any cochain complex that contains the -s themselves as cocycles?

]]>Oh, now I see, thanks! (however, I think that “ is not necessarily a direct product of with ” should be “ is not necessarily a direct product of with “, and “the relevant cocycle ” should be “the relevant cocycle “)

]]>The cocycle here (in the sense of the previous blog post) is not an -cocycle, but rather a -cocycle. The fact that is not necessarily a direct product of with is what produces the additional correction.

In a bit more detail: a typical element of is of the form for some , . This acts on via left multiplication using the law (1); in the notation of the previous post, the relevant cocycle is then given by

where the group action of on is given by projecting down to :

The cocycle equation

from the previous post should then correspond to the cocycle equation

from the current post.

]]>According the post Cohomology for dynamical systems, the phrase that “ is a cocycle” means that the coboundary of the function (cochain) is 0. The coboundary of an 1-cochain is

With the replacement ,, , , we get the coboundary of :

, so the cocycle condition would be

.

The term is not present here. So why is called the equation having this excess term a “cocycle condition”?

]]>Terry, I didn’t expect an example so remarkably simple! And thanks for the pointer.

]]>The simplest example is probably Cauchy’s functional equation:

http://en.wikipedia.org/wiki/Cauchy%27s_functional_equation

Any solution to this equation that is merely continuous (or even measurable), is automatically smooth (and even analytic).

Regarding the connection to the dichotomy, this is an example of what I call “rigidity” in these slides: http://www.math.ucla.edu/~tao/preprints/Slides/icmslides2.pdf

]]>I tried following the post, but got lost. But I find this comment fascinating, yet mysterious. Can you give a more “elementary” example of this phenomenon (of upgrading weak regularity to strong using algebraic structure)? Also what is its connection to the dichotomy (between structure and randomness)? If it helps, I am from a CS background.

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