13. Deformation Theor y of Pseudogrou p Structure s 13.
We want to show that cp, = if) . Thus we want to show that for any p with
*rr (p) = x we have
(2. 10) VepXx) ° P = 0(p) .
Both sides of (2. 10) a r e jets with the s a m e sourc e O and the s a m e targe t
p(x) . In o r d e r to establis h (2. 10) it suffices to show that for any v £Jn(T(]Vl))n
we have
(2-11) Vp)(x)*P*v = J/)(P)*V .
To do this , choose a £ £ T (D^OjM)) with T T ^ | = p ^ v . Then, by (2. 9),
J
1
M( X )*P* V
=
^P*V
= ^ ^ t = *#**(*)
Since (2. 8) holds , $(p)^ **# ## (4) = P* IT* 4 ~ v . Thus (2. 11) holds proving
the lemma .
To prove the proposition for k we ma y a s s u m e that it has bee n
establishe d for k - 1 . Let p
k
_
l
: J
k
.
L
( T ( M ) )
0
- J
k
_
2
( T ( M ) )
Q
be the
obvious projection map . It is then an immediat e consequence of the
definitions that for any 4 T (D
k
(0;M) ) ,
(\k-iMtfk-i =
Pk-i(*'
a
k»'
i
-
e
" Pk-i^kJ
=
^k
k
-i^
a
k-i
If (•nj^)* 1 = 0 then (2. 8) implie s that (^_
1
) # 0* £ = ° - Thus 0
is a fiber ma p with r e s p e c t to the fibration D, (0;M) D, ,(0;M) . It is
eas y to check that 0 induces a ma p on D, ,(0;M) p r e s e r v i n g 0^-1 anc *
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