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

TL;DR

This work proves that for sequences of almost-multiplicative maps $\rho_n$ into unitary groups, the $k$th Chern character $\mathrm{ch}_k(E_{\rho_n}^Y)$ vanishes for all $k\ge p$ when multiplicativity is measured in the unnormalized Schatten $p$-norm. The authors develop a differential-forms framework and construct almost-flat bundles $E_{\rho_n}^Y$ whose curvature forms become arbitrarily small in $p$-norm, forcing higher Chern characters to vanish. Consequently, higher even cohomology obstructions cannot obstruct stability in the unnormalized Schatten $p$-norm for $p$, $k$ with $k\ge p$, clarifying the limits of cohomological obstructions in this stability setting. The results yield corollaries about dimensional bounds, show that operator-norm stability does not always lift to Schatten $p$-norm stability, and illuminate the role of the $\gamma$-element and quasi-diagonality in relation to Dadarlat’s obstruction.

Abstract

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.

Theorems & Definitions (64)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Proposition 2.4
• proof