This question is a follow up of my previous question.
Q1: By having a list of homotopy groups and given dimension $n$, Is it possible to recover the topology?
According to WikiPedia: Topological spaces that are not homeomorphic can have the same homotopy groups.
Q2: Can anyone give an example of the above fact for smooth manifolds such that they have same dimension?
Q3: Can the homotopy groups determine the Euler characteristic?
A1: No
A2: An example can be: $M_1= S^{1} \times (-1,1)$ (i.e Cylinder) and $M_2$ will be the open mobius strip. One way to represent the Möbius strip as a subset of three-dimensional Euclidean space is using the parametrization:
$$x(u,v)= \left(1+\frac{v}{2} \cos \frac{u}{2}\right)\cos u$$
$$y(u,v)= \left(1+\frac{v}{2} \cos\frac{u}{2}\right)\sin u$$
$$z(u,v)= \frac{v}{2}\sin \frac{u}{2}$$
where $0 \le u< 2\pi$ and $-1 < v< 1$.
They are both Homotopy equivalent to $S^{1}$ so they have the same homotopy groups (Moebius band not homeomorphic to Cylinder.).
A3: They don't as there are spaces with the same homotopy groups but different homology groups(Spaces with equal homotopy groups but different homology groups?).
