к Jamch

property (В), if e is additive with respect to disjoint union of closed manifolds and for any two compact bounded differentiable manifolds Mq and M^ and diffeomorphisms cp дМсу-^дМ^ and \p ôMq-^ôMi,wq have

£ ( Mo u^Mi) = e(Mo u^Mi)

In particular, if s is defined for all bounded manifolds and e(Mo u^M^) = e(Mo) + e(Mi) for all diffeomorphisms cp дМо-^дМ^, then the restriction of e to closed manifolds has property (B) The Euler characteristic e m even dimensions provides an example of such an invariant, because e(Mo"u^Mi^") = e(Mo) + eiMi)-e{dMo) = e{Mo) + e{M^), since ôMq is a closed odd dimensional manifold

Our main result is

Theorem 1. (a) Let a have property (A) and define a^ = (т{Р2к{0) Then for any closed oriented manifold M" we have

^ ( ^ " ) = Л. .^.,4k

0 if мф0mod4

aj . TiM''^ ) if n = 4k

( b ) Let 8 have property (B) and define bj, = b{P2k{^) Then for any closed manifold M" we have

0 if n IS odd

„ fO if n IS

'^^ " ^^\Ke ( M'' ) If n = 2k

Proof Consider the oriented case If X is closed and we apply (A) to Mo = X, Ml = M2 = 0, we obtain a{X) = -ai-X) If Z and Y sltq closed, we get a{X + 7) + ö-( - X) + a( - У) = 0, thus a is additive with respect to disjoint union (The "-h" sign between manifolds indicates disjoint union) Now let M be bounded and X = M u^m -^ the usual "double" of M Putting Mq = M^ = M2 = M, we obtain from (A) (t(X) + ö-(-X)4-ö-(X) = (t(X) = 0, and since cr(Mo u -Ml) does not depend on the choice of cp, we see that a vanishes for all closed manifolds of the form Mu^ -M As a special case, we have

( 1 a) // the closed oriented manifold X is fibred over a positive dimensional sphere, then (т{Х) = 0

Here and m the following we use the word "fibred" in the sense of "being the total space of a locally trivial differentiable fibration "

Now consider the unoriented case We cut S^ into two pairs of intervals Then there are two essentially different ways cp and ip to re-attach these interval pairs to a closed manifold, giving S^ in one case and the disjoint union of two copies of S^ m the other Thus (B) imphes e{S') = s{S'+ S') = eiS') + 8{S% hence e{S^) = 0 The same argument shows that in fact s vanishes for any closed manifold which is fibred over S^ We now generalize this to

( 1 b) If the closed manifold X is fibred over an odd dimensional sphere, then

8 { Х ) = 0