Manuscript III@2 · III.tex · proposition · 82dad3e5

Proposition: Poisson compensation

The unchanged free-product obstruction › Spatial Poisson compensation · lines 643–664 · proof included or linked

III@2#prop:Poisson

For compact smooth $ξ$ ,

w_{P}^{μ} (h + K ξ) = w_{P}^{μ} (h) - ξ^{μ} .

Proof. Directly,

A_{i} [K ξ] = Δ ξ_{i}, B_{P} [K ξ] = Δ ξ_{0} .

Therefore

w_{P}^{i} [K ξ] = - ξ_{i} = - ξ^{i}, w_{P}^{0} [K ξ] = ξ_{0} = - ξ^{0} .

Use linearity.

Displayed formulas in this passage

unlabeled · line 646
$w_{P}^{μ} (h + K ξ) = w_{P}^{μ} (h) - ξ^{μ} .$
unlabeled · line 653
$A_{i} [K ξ] = Δ ξ_{i}, B_{P} [K ξ] = Δ ξ_{0} .$
unlabeled · line 658
$w_{P}^{i} [K ξ] = - ξ_{i} = - ξ^{i}, w_{P}^{0} [K ξ] = ξ_{0} = - ξ^{0} .$