What is needed to define the index of a vector field? All of the definitions that I have seen for the index of a vector field $X$ on $\mathbb{R}^n$ at some singularity $p$ go as follows: first let $D$ be a contractible domain on which $p$ is the only zero of $X$. Then consider the Gauss map $\partial D \to S^{n-1}$ given by $q \mapsto \frac{X(q)}{\lVert X(q) \rVert}$. The degree of this map is the index of $X$ at $p$.
However, if $X$ is a vector field on a more general manifold $M$ it's not clear to me how to extend this definition, since it seems to require some sort of connection or metric to compare values of the field at different points. Of course, it happens that the previous definition makes sense after a choice of coordinates, and the result is of course independent of such a choice. This seems to indicate there should be a more satisfying way of defining the index, using cohomology in some way.
Lemma 11.17 of Madsen and Tornehave's "From Calculus to Cohomology" seems to point in that direction, but it still identifies a vector field with a function from $\mathbb{R}^n$ to $\mathbb{R}^n$.
 A: Let $n$ be a positive integer, $V$ a real $n$-dimensional vector space, and $V^{\times}$ the set of non-zero vectors in $V$. Let $\sim$ denote the equivalence relation on $V^{\times}$ whose classes are rays through the origin, i.e., $v \sim v'$ if and only if $v' = cv$ for some positive real $V$. Finally, define the $(n-1)$-sphere to be the quotient $V^{\times}/\sim$, in analogy to the usual construction of a real projective space.
One difference with the usual picture of a sphere is, this sphere does not naturally embed in $V^{\times}$. Analogously, a quotient vector space $V/W$ does not naturally embed in $V$, though picking an inner product on $V$ does allow us to identify $V/W$ with the orthogonal complement of $W$.
This construction can be applied fibrewise to an arbitrary continuous vector bundle over a topological manifold. Particularly, if $M$ is a topological $n$-manifold, $p$ a point of $M$, and $X$ a vector field (perhaps viewed as a suitable section of the tangent microbundle of $M$) with an isolated zero at $p$, then $X$ induces a continuous self-mapping of the sphere by restricting to the boundary of a ball neighborhood of $p$ and taking the direct limit under inclusion. The degree of a mapping from the sphere to itself can be defined in the usual sense of induced mappings on homology.
Caveats: I am not a topologist; there may be better, and/or well-known, ways to define the degree of an isolated zero of a continuous vector field. Taking a direct limit over inclusion seems like a lot of contortion to avoid using a local trivialization to compare values of $X$ at different points, and the only way I see to verify the direct limit exists is to use local coordinates, which may be philosophically antithetical to the question.
