I was advised to post my question here. A colleague suggested a proof of a fact which I have hard time to believe. Since I am not a topologist by training I wonder if this can be true in such a generality.
Consider any finite CW-complex structure on the -dimensional sphere $^$. Let $$ be $^$ with all strata of codimension at least 2 removed, i.e. we keep just and (−1)-dimensional strata of the complex under consideration.
Claim. K is contractible to a graph (the dual graph of the preserved strata). Thus it is a $K(\pi_1, 1)$- space where $\pi_1$ is a free group.
The claim is trivially true for d=2, but looks suspicious for higher d. Also if it is true then it might work for more general CW-complexes as well.
Best regards, Boris Shapiro/Stockholm University/