When does the diagonal cohomology class of a non-compact oriented manifold vanish? Let $M$ be a non-compact, connected and oriented topological $d$-manifold without boundary. My understanding is that there are two (equivalent) ways of defining the diagonal class $\delta_M \in H^{d}(M \times M; \mathbb{Z})$:

*

*Consider the fundamental class $[M] \in H^{lf}_d(M;\mathbb{Z})$ in locally finite (=Borel-Moore) homology. The diagonal map $M \rightarrow M \times M$ is proper, hence induces a map in locally finite homology. From here we get a class
$$[M] \mapsto \Delta_M \in H^{lf}_d(M \times M;\mathbb{Z})\,.$$
By Poincaré duality in the connected oriented 2d-manifold $M \times M$ we get a dual class $\delta_M \in H^d(M \times M;\mathbb{Z})$, that is, $\delta_M$ is the unique class with $\delta_M \cap [M \times M] = \Delta_M$.


*Take $\delta_M$ as the image of 1 under the Gysin map $\mathbb{Z} \cong H^0(M;\mathbb{Z}) \rightarrow H^d(M \times M;\mathbb{Z})$ induced by the diagonal map.
Question: Is there a characterization of the class of manifolds $\mathcal{C} := \{M: \delta_M = 0\}$? One obvious sufficient condition for $M \in \mathcal{C}$ is that $M$ has no cohomology to begin with, e.g. is contractible. Second, $\delta_{M \times N} = \delta_M \times \delta_N$ under the cohomological cross product so if $M \in \mathcal{C}$ then $M \times N \in \mathcal{C}$ for any $N$. Third, for any open subset $U \subseteq M \in \mathcal{C}$ we have $U \in \mathcal{C}$ because in general $\delta_M$ pulls back to $\delta_U$. Are there other easy to check sufficient or necessary conditions?
 A: There are some closely related notions appearing the first part of "Espaces de configuration généralisés -
Espaces topologiques i-acycliques -
Suites spectrales basiques" by Arabia.
Note first that an equivalent condition is that the Gysin map vanishes in all degrees, i.e. $H^\ast(M) \to H^{\ast+d}(M \times M)$ is zero. This is because the Gysin image of a cohomology class $\alpha$ is the cup-product of $\delta_M$ with the pullback of $\alpha$ to one of the factors.
Arabia considers the linear dual condition: he defines a topological space to be $\cup$-acyclic if the cup product in compact support cohomology is identically zero. On an oriented manifold this just means then that the intersection product on homology $H_{\ast+d}(M \times M) \to H_\ast(M)$ is zero. With field coefficients this is equivalent to your condition.
Proposition 1.2.4 of his paper gives some more characterizations of this property, in particular that on an oriented manifold it is equivalent to asking that $H^\ast_c(X)\to H^\ast(X)$ vanishes. He also writes down exactly the same examples as you do.
