in the generalized lifting-extension problem, then it can be proved that $[X,Y]_{EG}^\emptyset$ is in a bijective correspondence with all equivariant maps $X\to Y$ up to equivariant homotopy. This is how we obtain Theorem \[theorem:equivariant\].[^6]

3.  (Extending sections in a vector bundle.) Let $G=\{e\}$, and let $\varphi{\colon}Y\to B$ be the inclusion ${BSO}({{n}}-k){\rightarrow}{BSO}({{n}})$, where $BSO({{n}})$ is the classifying space of the special orthogonal group $SO({{n}})$. Then the commutative square in the generalized lifting-extension problem is essentially an oriented vector bundle of dimension ${{n}}$ over $X$ together with $k$ linearly independent vector fields over $A$. The existence of a solution is then equivalent to the existence of linearly independent continuations of these vector fields to the whole of $X$. We remark that in order to apply our theorem to this situation, a finite simplicial model of the classifying space $BSO({{n}})$ would have to be constructed. As far as we know, this has not been carried out yet.

    We briefly remark that for non-oriented bundles, it is possible to pass to certain two-fold “orientation” coverings and reduce the problem to the one for oriented bundles but with a further ${\mathbb Z}_2$-equivariance constraint.

The main theorem {#the-main-theorem .unnumbered}
----------------

Now we are ready to state 