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How can one generalize the Euler characteristic to non-compact manifolds?
Furthermore, is there a way to generalize the notion of an intersection number to non-compact manifolds, so that one could use the definition of the Euler characteristic as the self-intersection number of the diagonal?

References for further reading would also be appreciated.

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The Euler characteristic is defined for any topological space with finitely many nonzero homology or cohomology groups (with coefficients in a field), all of which are finite-dimensional; it's just the alternating sums of the dimensions. This is in particular the case for "reasonable" noncompact manifolds.

However, arguably this is the "wrong" answer for noncompact spaces, and one might instead prefer the compactly supported Euler characteristic, which is defined using Borel-Moore homology or equivalently compactly supported cohomology.

For example, the Euler characteristic of $\mathbb{R}^n$ is $1$, but the compactly supported Euler characteristic is $(-1)^n$. (In particular it is not a homotopy invariant.) Note that the Euler characteristic of the sphere, which is the one-point compactification of $\mathbb{R}^n$, is $1 + (-1)^n$; one reason to prefer the compactly supported Euler characteristic is that it has better additivity properties.

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  • $\begingroup$ There should be a notion of intersection theory on noncompact manifolds but I'm not sure what the precise statements are. For starters, noncompact manifolds don't have a fundamental class living in cohomology but they do have a fundamental class living in compactly supported cohomology. Poincare duality in this setting also involves compactly supported cohomology. $\endgroup$ – Qiaochu Yuan May 14 '14 at 19:05

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