On The Functors CW_{A} and P_{A}.
Wojciech Chacholski
I am looking at the relation between Bousfield's localization
functor P_{A} and Dror Frajoun's colocalization functor CW_{A}.
I am studying the question: to what extent the following sequence
is exact:
Spaces --P_{A}--> Spaces --CW_{A}--> Spaces --P_{A}--> Spaces
The image of P_{A} is equal to the kernel of CW_{A}.
The correlation between the kernel of P_{A} and the image of CW_{A}
is more complicated. I proved that The kernel of P_{A} is the
closure of the image of CW_{A} under taking extensions by fibrations.
In the paper I am giving algorithms to construct the functor
CW_{A} out of P_{A} and vice versa. I am using these algorithms
to show that S^{n} is in the image of CW_{\Omega S^{n+1}} if and
only if n=1,3,7 and that for every n S^{n} is in the kernel of
P_{\Omega S^{n+1}}.