Do pseudomanifolds have an intersection form? They have a volume cycle so it seems like you can use this to define a pairing on $H^{middle}(X,\mathbb{Z})$, however I've only see references to the intersection form on $IH(X)$.
One reason I'm interested is I want to know whether non-unimodular quadratic forms can make an appearance on pseudomanifolds without a Poincare duality. Are there any examples of this in 4 dimensions?