Euler class in the non-compact case Does anyone have a reference for:
The Euler-class for an open non-compact
manifold possibly with twisted coefficients (if the
group action on the manifold does not preserve
orientation) and/or for a compactification
e.g. the one point compactification
jim 
 A: For the untwisted case see Dold's "Lectures on algebraic topology" section VIII.11. If $N$ is a oriented topological submanifold of an oriented manifold $M$ of codimension $k$, then one looks at the Thom class in $H^k(M, M-N)$ and then restricts it to $H^k(N)$ to get the Euler class. Compactness of the submanifold $N$ is never needed.
A: There is one version of Euler class for oriented  vector bundles on non-compact manifolds, the so called relative Euler class . It  requires that the vector bundle admits a section which does not vanish  outside a compact set.  The relative  Euler class  is then an element of the cohomology with compact supports, and as such, it depends on the choice of the section that  is  nontrivial outside that compact set.  
Formally, if $E\to M$ is an oriented vector bundle with Thom class $\tau$ and $s:M\to E$ is a section that does not vanish outside a compact set, then the relative  Euler class is
$$\boldsymbol{e}(E, s):=s^*\tau(E)\in H^r_c(M), $$
$r$ being the (real) rank of $E$.  $\newcommand{\be}{\boldsymbol{e}}$ The class $\be(E,s)$ depends only the homotopy class of $s$  in the space of sections nontrivial outside a compact set.
If $M$ happens to be oriented, then $\be(E,s)$ is the  Poincare dual of the cycle determined by the zero set of $s$.
Here is a good example to think about. Suppose that $L\to D$ is the trivial complex line bundle over the open unit disk in the plane.   Suppose  $s(z)=z^k$,  $k\geq 0$. Then
$$\be(L,s)\in H^2_c(D)= H^2(D,\partial D)$$
and
$$\langle \be(L,s), [D,\partial D]\rangle =k. $$
A: A version of the Euler class for oriented noncompact manifolds appears in the paper "Fixed-point theories on noncompact manifolds" by Shmuel Weinberger (you need a Riemannian metric of bounded geometry, I think): http://math.uchicago.edu/~shmuel/fpt.pdf The setup is similar to the one in compact case, but you need to have uniform bounds on the vector fields. The non-orientable case should be similar. 
