Vanishing of Topological Invariants For Unnormalized Schatten $p$ multiplicative Maps

A result of Dadarlat shows that nonzero even rational cohomology obstructs the matricial stability of many discrete groups. In the author's previous work, 2-cohomology is used to argue that certain groups are not stable in unnormalized Schatten $p$-norms for $p > 1$, although 2-cohomology is known not to obstruct stability in the unnormalized 2-norm in general. The main result of this paper demonstrates that we should not expect $2k$ cohomology to obstruct in the unnormalized Schatten $p$-norm for $p\le k$, because the invariant in Dadarlat's argument vanishes for maps that are asymptotically multiplicative in the Schatten $p$-norm.