Manuscript XVI-I@1 · XVI-I.tex · section · a7046a8d

The compressed route

Root attachment: maps, wavefront test, and properness › The compressed route · lines 1347–1379

XVI-I@1#eq:attachment, XVI-I@1#eq:root-condition

Let

X = M_{y} \times M_{w} \times M_{x},

and define the projections

p_{B} (y, w, x) = (y, x), p_{U} (y, w, x) = (y, w), π (y, w, x) = (w, x) .

They are submersions. The proposed compressed graph distribution is

G [h] = A \sum (p_{B}^{*} B_{A}) (p_{U}^{*} U ˉ_{A} [h]), R [h] = π_{*} G [h] .

For a general root, the product would require

WF (p_{B}^{*} B_{A}) \cap (- WF (p_{U}^{*} U ˉ_{A} [h])) = \emptyset

for every component, followed by properness of $π$ on the product support.

Displayed formulas in this passage

XVI-I@1#eq:attachment · line 1363
$G [h] = A \sum (p_{B}^{*} B_{A}) (p_{U}^{*} U ˉ_{A} [h]), R [h] = π_{*} G [h] .$
XVI-I@1#eq:root-condition · line 1372
$WF (p_{B}^{*} B_{A}) \cap (- WF (p_{U}^{*} U ˉ_{A} [h])) = \emptyset$
unlabeled · line 1350
$X = M_{y} \times M_{w} \times M_{x},$
unlabeled · line 1354
$p_{B} (y, w, x) = (y, x), p_{U} (y, w, x) = (y, w), π (y, w, x) = (w, x) .$