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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