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

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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

$$\langle \be(L,s), [D,\partial D]\rangle =k.$$

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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.