6
PRELIMINARIES
respect to the ith. variable) of F(6iK). Hence we have homotopy exact sequences
7Tr(diK) —• 7Tr(6iK) - 7Tr(K) -+ 7Tr-i(diK) - . . .
(indeed, we could map in any space, not just spheres).
We can also define bordism groups and obtain corresponding exact sequences.
We adopt the philosophy of [W13]. We first define the boundary (n + l)-ad,
dK, of K by
dK(a) = \J{K(p):(3ca,/3^a}.
We call an (n+l)-ad M a manifold (n+l)-ad if for each a C {1, 2,... , n}, M(a)
is a manifold with boundary dM(a). We obtain correspondingly PL manifold
(n + l)-ads and smooth manifold (n + l)-ads: in the latter case, we usually
assume that each M(a) and M(/3) meet transversely at M(a D /?), so we have
a 'variete a bord anguleux' in the sense of Cerf [C12] or Douady [D3].
Note that the contracted n-ad cM is again a manifold n-ad. In the case when
|#
n
_iM| and |d
n
M| are disjoint, we regard either M or cM as a cobordism
between the n-ads dn-\M and dnM: the latter are effectively (n l)-ads since
9n_iM = an6n(dn-iM); similarly for dnM.
For cobordism purposes we will always assume \M\ (and hence each M(a))
compact. One can study cobordism of manifold (n + l)-ads in general: this
study is meaningful only if we impose various restrictions on the manifolds
and cobordisms considered, with stronger restrictions on each \diM\ than on
\M\. For example we have the plain bordism groups of an (n + l)-ad, K.
Consider maps 0 : M K with M a (smooth, compact) manifold (n 4- 1)-
ad, dim \M\ = m. Using disjoint unions of M gives the set of such (M, /) the
structure of an abelian monoid. We set (M, 0) ~ 0 if there is a manifold (n + 2)-
ad A^ with M = 9n+iA^, and an extension of 0 to a map ip : ./V s
n +
i X of
(n + 2)-ads. Then set (Mi,0i) - (M2,02) if (M^^i) + (M2,02) ~ 0 (cf. the
definition of cobordism above). The usual glueing argument (note the utility
here of our corners) shows that this is an equivalence relation; it is evidently
compatible with addition. We thus obtain an abelian group J/^^K). Clearly
jV^iV) = jVraipiV) \ the inclusions d{K c ^ K , OibiK C K, and restriction
define sequences
which are easily seen to be exact. Since excision holds for unoriented bordism,
it is easily seen that J/^K) = J/^i^K^ \dK\). This remark will not, however,
apply to all the generalisations which we will need.
We introduce a convention for the oriented case which will be useful later on.
Observe that for manifold (n + l)-ads in general, dimM(a) \a\ is independent
of a (we ignore cases M(a) 0 here). We denote this number by dimM{ } :
of course if M{ } is empty, this is a convention. Suppose \M\ orientable: then
so is d\M\ = U {M(a) : |a| = n 1}; by downward induction on |a|, we deduce
that all M{a) are then orientable. More precisely, an orientation of \M\ induces
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