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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


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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.

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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

$$\be(L,s)\in H^2_c(D)= H^2(D,\partial D)$$


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

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Two questions: What is the reference for this? What if the bundle is relatively orientable? i.e. the real line bundle $det(E^*)\otimes det(TM)$ is trivial. In this case, for a transverse section $s$, the zero set is of $s$ is a compact oriented submanifold of $M$; thus, defines an element in $H_{dim M-rank E}(M)$. However, we dont have Poincare duality to get some cohomology class. – Mohammad F. Tehrani Feb 24 at 22:14
The Euler class, as a cohomology class is well defined for any oriented vector bundle over any compact $CW$ complex. In particular, the base of the bundle need not be a manifold. The book of Milnor and Stasheff on characteristic classes is a good place to consult. – Liviu Nicolaescu Feb 25 at 1:01
I particularly meant the non-compact case and most importantly, not necessarily orientable but just relatively orientable (where the zero set of a compactly supported transverse section would still be orientable (oriented)). – Mohammad F. Tehrani Feb 25 at 3:06

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): 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.

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