Suppose that we are given a topological space $X$: assume for simplicity that $X$ is compact we want to adress the following question:

Is it true that one can find a manifold $M$ which is homotopy equivalent to $X$?

**Necessary condition** $X$ must satisfy Poincare duality, namely there must be a class in the top homology of $X$ such that the cap product with this class induces isomorphism between homology and cohomology. As one could expect this is not enough. So let us assume that $X$ indeed satisfy Poincare duality. Then one can associate to $X$ the so called *Spivak normal fibration*. This fibration is classified by the map $f:X \to BG$ where $G$ is a space of self-homotopy equivalences of the sphere.

**Second assumption**: Let us assume that $X$ is simply connected. Then one can find a manifold homotopy equivalent to $X$ which is:

-smooth iff the map $f$ lifts to a map $X \to BO$

-piecewise linear iff the map $f$ lifts to a map $X \to BPL$

-topological iff the map $f$ lifts to a map $X \to BTop$.

(Here as far as my knowledge goes, $Top$ is the space of homoeomorphisms of the sphere, $PL$ the space of piecewise linear homeomorphisms of the sphere and so on-but please correct me if I'm wrong).
Problem of lifting maps leads to the obstruction theory which gives classes in cohomology $H^{n+1}(X,\pi_n(F))$ where $F$ is a fiber of our fibration which we would like to lift (in our case $F$ is equal to $G/O,G/PL,G/Top$ respectively).

Question 1It is general knowledge of the obstruction theory that it works better in the simply connected case: in the non simply connected case one has to deal with local coefficients. In our problem of finding a manifold in the given homotopy type in the non simply connected case there is a further obstruction: is it right to think that the presence of this obstruction is due to the fact that one has to work with local coefficients?

Question 2What is known about the homotopy groups of the fibers $G/O$ (resp. $G/PL$, $G/Top$)? Is it somehow possible to interpret the various theorems of the form ,,up to some dimension/in some dimension two notions of manifolds (e.g. smooth, PL, Top) coincide'' in this language, i.e. as vanishing of the homotopy groups of the fiber up to some dimension?

Question 3Due to the theorem of Sullivan every manifold of dimension different than $4$ has unique Lipschitz atlas: how does it fit into this picture?

My question is very broad, it is rather some kind of a big picture: nevertheless I hope that it would be interesting to someone who is less familiar with this theory.

compact. In fact, there are plenty of open manifolds that donotsatisfy Poincaré duality. $\endgroup$ – Francesco Polizzi Jul 23 '18 at 7:24