Can one prove that orientability(of a manifold)is a topological property without using algebraic topology? That is, using a combination of general topology,linear algebra,and topological groups(such as Lie groups-notably the orthogonal group $O(n)$). The basic idea I have would be to show that the group actions of orientation reversing orthogonal linear transformations(elements of $O(n)$ with $Det=-1$) on a vector field defined on an orientable surface are not homotopic to the actions by an orientation preserving transformation(elements of $O(n)$ with $Det=+1$). An example I could think of are the actions of $I_3$ and $-I_3$(identity and antipodal map)on a normal vector field defined on the unit sphere $S^2$ embedded in $\mathbb R^3$. To reverse the direction of a normal vector on S2 involves rotating it about an orthogonal axis tangent to $S^2$ and passing it through the surface.