By H. Jacquet, R. P. Langlands
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THOM introduced the spaces above in order to convert D,. into a stable homotopy group. We give now a bare outline of this noted theorem. 1) Thom. +k(MSO(k)), and m.. +k(MO(k)). The proof is long [40, 22]. Herewe merely define the isomorphisms. Suppose that M" is a closed oriented n-manifold. Embed M" in SnH via the Whitney embedding theorem. Denote by ~ : A -+ M" the normal cell bundle to M" in 5"+k. Assuming Sn+k oriented, we may assume the tangent bundle p, to Sn+k oriented; moreover the tangent bundle -r to Mn is oriented.
1) this is a monomorphism with image H .. (X, A; Z) ® QllcH.. o is a permanent cycle, so is every element of H7J(X,A ;Z) ® DaC H7l(X,A; Dq)· Now H7l(X, A; Z) ® Da is a direct summand of E;,q. the other summand TM being isomorphic to Tor(H'J)-l (X, A; Z), Da)· Since Da has no odd 42 II. Computation of the bordism groups torsion, Tv,a consists of 2-torsion only. 1), d 2 carries E~,q onto an odd torsion group, and since T v. a) = 0. Thus d2 = 0. As we continue through the spectral sequence, it is seen to be trivial.
Any Stiefel-Whitney dass of the product M: X vm is of form W; = 1 ® v1 + terms involving Stiefel-Whitney classes of M:. From dimensional considerations we see that W;, ... W;k(f;n)*cn,i = lt cn, i ® V;, ••• V;k' Thus N N (W;, ... W;k(l;n)*cn,i, an X am) N = = (I* (cn,i), an) (v;, ... V;k' am) (v;, ... = i, the Whitney numbers of the product associated with c"·; all vanish. Now suppose there is an expression Em,i [M:-m, 1;] 2 [V:'J 2 = 0. All the Whitney numbers must vanish. We show inductively that [V:'J 2 = 0 for all m, i.