LECTURE 5. MULTI-VALUED PERTURBATIONS 211Figure 20. Local slices for the G-action on Map( S^2 , M)Corollary 5.5. Let u E Map(S^2 , M) be a smooth function with finite isotropy sub-
group Gu of order m. Then for every rJ E D^0 •^1 (S^2 , u*T M) there exists a weighted
multi-valued G -section (I',>.) which satisfies the ''finiteness", "conformality", "en-
ergy", and "local structure" ax.ioms, as well as
A '( U,T) ) = # {cp E Gu : cp*TJ = TJ}.
mProof. Let p : uk,p ----+ 2Hor~·P x G be the local slice of Lemma 5.3, fix an element
v E Uk,p, and write
p( v) = { (6, 7/J1),. · ·, ((m, 7/Jm)} ·
Then v = expu((i) o 7/Ji for all i. Let u(() : uTM----+ expu(()TM be given by
parallel transport as in the footnote on page 209. Then
for each i. Now choose a smooth cutoff function f3: JR----+ [O, 1] which is equal to 1
in [-1/2, 1/2] and equal to zero outside [-1, l ]. Then define /3 0 ,p: Horu----+ [O, 1] by
/3,.,(<) ~ /3 (JS' ( (£-'l<I' + £)'
1
' + '~" (£-
2
1 \7(< O 'l')I' +£)Vi') Wu) ·This is a smooth Gu-invariant cutoff function on Hor;;P vanishing outside the €-
neighbourhood of zero. Next define
I'(v) = {(1,.. .,(m}, .A( v, () = #{ i : ( i = (})
mwhere ( i = /3 0 ,p((i)(u((i)TJ) o d7/Ji E Eu for i = 1, ... , m. This perturbation has the
required properties. In particular, note that (i = ( i' if and only if cp*TJ = rJ where
cp = 7/Ji' o 7/Ji -l E Gu. Hence the number of distinct branches is the quotient m/mTJ,
where mTJ = #{cp E Gu : cp*TJ = rJ}, and two distinct branches agree precisely on
the zero set of the cutoff function /3 0 ,p (which is connected and contained in the
closure of its interior). D
So far we have constructed a perturbation which satisfies the ''finiteness", "con-
formality", "energy", and "local structure" axioms, but not the "transversality" and
''free" axioms. To find a perturbation which also satisfies the "transversality" axiom
it is useful to construct a sufficiently large family of perturbations, parametrized
by a separable Hilbert space H, by superposition. This is possible because of the