Manuscript XIII-D@1 · XIII-D.tex · lemma · bab69024

Lemma: The detecting parameter is a commutator

D5: the copied witness on a parameter commutator · lines 998–1019 · proof included or linked

XIII-D@1#lem:bracket, XIII-D@1#eq:theta

The parameters (eq:bracket-parameters) satisfy

[ξ, η] = h (t)^{2} ζ (x)^{2} \partial_{t} =: θ .

Proof. Both vector fields have only a temporal component. Their spatial cutoff is independent of time. Thus the temporal component of their bracket is

h ζ \partial_{t} ((t - t_{*}) h ζ) - (t - t_{*}) h ζ \partial_{t} (h ζ) = h^{2} ζ^{2} .

There are no spatial components.

Displayed formulas in this passage

XIII-D@1#eq:theta · line 1001
$[ξ, η] = h (t)^{2} ζ (x)^{2} \partial_{t} =: θ .$
unlabeled · line 1013
$h ζ \partial_{t} ((t - t_{*}) h ζ) - (t - t_{*}) h ζ \partial_{t} (h ζ) = h^{2} ζ^{2} .$

Nearby in this section

section: D5: the copied witness on a parameter commutator
section: D5: the copied witness on a parameter commutator
theorem: Theorem: Cocycle obstruction for the frozen class
corollary: Corollary: What the witness does and does not cancel
section: D5: the copied witness on a parameter commutator