In #9, if f has no fix point, then if gives a free action by the group Z/2Z on R^n. Then as Infrared points out, we get an equivalence relation by setting x ≈ f(x) and the quotient map to the set of equivalence classes, R^n-->X = R^n/{x≈f(x)} is actually a covering space map with fibers of degree 2. Thus X is a manifold of dimension n, with fundamental group Z/2Z, and contractible universal covering R^n. Since maps of simply connected spaces lift to covering spaces, and homotopies push down, it follows that all homotopy groups of X in degree ≥ 2 are trivial, since this holds for R^n.
Now this reminds us of real projective space P. I.e. P^2 has fundamental group Z/2Z, but has higher homotopy groups, since it is covered by S^2 which has them. but P^3 also has π1 = Z/2Z, and no π2, since S^3 has none. So if we crank up to P^infinity, we kill all πk, for k ≥2, while keeping π1 = Z/2Z. Then we ask whether this would force our X above to be homotopy equivalent to P^infinity. Indeed it does, as we can deduce from Whitehead's theorem, provided we find a map from P^infinity to our X that induces these isomorphisms on homotopy groups.
'This follows from the construction of P^infinity by adding cells one at a time to P^2. First we map S^1 to our X as a representative of the generator of π1. Then since twice that map is homotopic to zero, we can extend twice that map to the disc, but since P^2 is constructed by gluing a disc to S^1 by twice the identity, this means we can extend our map of S^1 to X, to a map of P^2 to X. Now if we map S^2 onto P^2 by the antipodal map, we get a map of S^2 into X, identifying antipodal points, and since X has no π2, the map extends to a map of the solid ball into X, also identifying antipodal points of the boundary sphere, hence defining a map of P^3 into X. Continuing to infinity, I hope we get a map of P^infinity to our X, inducing isomorphisms of all homotopy groups, hence by Whitehead, defining a homotopy equivalence of P^infinity with our X. But this is a problem, since P^infinity is infinite dimensional and our Xn is a finite quotient of an n manifold. So can we make this a contradiction?
One way is to compute the homology groups of these spaces. An n manifold has them only up to at most dimension n, and we claim that P^infinity has them in infinitely many degrees, but why?
I guess I don't know this homology theory, so need to learn it before answering further. but people usually compute homology by a maier vietoris sequence.
ok there is an explanation in Greenberg chapter 19, of how to compute the change in homology from adding an n cell to a space, and it gives projective space as an example, and in the case of real projective space with Z/2Z coefficients, every time you add a new n-cell you get a new copy of Z/2Z in n dimensional homology . Thus for P^infinity you have Z/2Z as homology in every dimension. In particular it is not homotopy equivalent to any finite dimensional (connected) manifold. qed. X does not exist.