Explanation of four 6000-level graduate exercises:

*Represent the isomorphism classes of group extensions 1 ⟶ ***Z**^{n} ⟶ Γ ⟶ C_{2} ⟶ 1. First step in this algebra problem is a paper of I Reiner (1957). Second step is calculation of the group cohomology H^{2}(C_{2};**Z**^{n}), whose origin is O Hölder's factorset (1893). Third step is construction of models.

*Compute the rational cohomology ring of the Eilenberg–Maclane space K(***Z**,n). First done by J P Serre (1951) with spectral sequences. Applied to show the famous result: the stable homotopy groups of spheres are finite.*Compute the homotopy set [X,S*^{n}] and for n odd [X,**R**P^{n}] in terms of basic invariants, where X is an n-dimensional CW complex. Application of 1940s and 1950s obstruction theory, respectively.*Define the obstructions to lifting a map X ⟶ B to X ⟶ E, where X is a CW complex and E ⟶ B is a fibration with simple fiber F.* Standard extension of 1950s obstruction theory with the use of twisted coefficients.