LECTURE 5. MULTI-VALUED PERTURBATIONS 213This space carries a natural rational label M(A; J, f) ~ Q : v r-t A(v) =
A( v, 8 J ( v)). It can be expressed as a union of countably many branches
Mi=M(A;J,1i)={vEUi: v*[S^2 ]=A,81(v)=1i(v)}.
In other worde, Mi is the zero set of the section 81 - Ii· The "transversality"
axiom asserts that this section is transverse to the zero section of £ and it followsfrom the implicit function theorem and the "local structure" axiom that Mi is a
smooth manifold of dimension
dim Mi = index Di,v = 2n+ 2c1(A).
Here we abbreviate D i,v = D81(v) - D 1i (v) : TvBk,p ~ £.t;-^1 ,P. This operator isthe vertical differential of the section 8 J - Ii at v (see the footnote on page 209).
The "local structure" axiom guarantees that this operator is Fredholm and its index
agrees with that of D v = DfJ 1 (v). In summary, the moduli space M = M(A; J, f)
is a union of countably many smooth manifolds, called the branches of M. For a
generic perturbation r it is a branched manifold in the following sense.
Definition 5.6. A branched m-manifold is a pair (M, A), consisting of a Haus-
dorff topological space M with a countable basis and a function A : M ~ Q, together
with a countable collection of triples {(Mi, Ai, <pi)}iEJ and a decomposition of the
index set I= LJi I(j) into finite sets such that the following holds.
(i) The sets M(j) = LJiEI(j) Mi form an open cover of M, for every i E J(j),Mi is closed relative to M(j) and, for all i, i' EI,
intM;' (Min Mi')= intM; (Min M i').
The Ai are positive rational numbers such that
x E M(j)
iEl(j)
xEM;
(ii) For every i E I the map 'Pi : Mi ~ JR.m is a homeomorphism onto an open
subset of JR.m. For every pair i, i' E I the transition map'Pi' 0 'Pi -l : 'Pi(intM; (Min Mi')) ~ <pi'(intM;, (Min Mi'))
is smooth.(iii) For every x EM there exists a continuous function Bx : M ~ [O, l] such
that Bx(x) = 0, Bx(Y) > 0 fo r y =/. x, and Bx o 'Pi-l : 'Pi(Mi ) ~ [O, l] is
smooth for every i.
A branched m-manifold with boundary is defined similarly. In this case the
charts 'Pi : Mi ~ lH!m are homeomorphisms onto open subsets of the upper half
space lHim = { x E JR.m : Xm 2: 0}, we define 8 Mi = 'Pi -l ( 81Hlm), and assume thatM i' n 8Mi c 8Mi'
for all i, i' E J. The boundary of M is then defined by