I have just read an article on " Orientation of manifolds" by Matthias Kreck ( http://www.map.mpim-bonn.mpg.de/Orientation_of_manifolds ) but I got stuck at one place. In the section " Reformulation in terms of local homological orientations " he consider a finite dimensional vector space $V$ equipped with an equivalence class of basis $ v_1 , v_2 , ...... , v_n $, then he defines a $n$-simplex with $0$ as an interior which turns out to be a generator of $H_n(V, V- 0)$.
My question: Suppose $ V=R^n$ .Then how does the vector space orientation implies homological orientation. If I take an " arbitrary basis" for $R^n$ and construct the $n$-simplex as he does which gives a generator of $H_n(R^n , R^n-0)$ then also local orientation has been defined. So where does the vector space orientation need here?
You are right, every basis induces also a homological orientation. He just wants to define a well-defined bijection $\{$vector space orientation$\}\leftrightarrow \{$homological orientation$\}$. So he starts with an equivalence class of bases.
Note that both sets contain $2$ elements, on the right there are the two equivalence classes of bases (Recall that the two equivalence classes correspond exactly to the two connected components of $GL_n(V)$). On the left we have the consistent choice of generators for $H_n(V,V-x)$ and every $x$ and its opposite generators.
So what you should check is that the map Matthias Kreck constructs is well-defined, i.e. equivalent bases are mapped to the same generators in homology.