Non - Orientable Surfaces in Orientable 3-Manifolds

87

Now suppose n +1 is not a power of two. Then

n = Y,aiT; a, = 0,1

and there is some integer /c^O with a^, = 0 and a^^+l = l. Then the binomial coefficient

[ 2^ )^^ (mod2) which imphes that

By the Wu formulae [4] and the assumption that the Stiefel-Whitney classes of M vanish, it follows that a" = 0 for all aeH^{M;Z2\ which proves the last statement. (This proof of the last statement was pointed out to us by E. Thomas.)

Now the first statement of the proposition is trivial unless n+1 is a power of two. Since addition in the cobordism group corresponds to addition of Stiefel-Whitney numbers, it suffices to show that a -^ a" is a homomorphism of H^ (M", Z2) -^ Я" (M" ; Z2) ^ Z2. However, if n +1 is a power of 2 then (a + jS)" - ^ a' ß" ' \ Thus

and again the Wu formulae and the assumption on the Stiefel-Whitney classes show that this is zero.

3 . Applications to 3-Maiiifolds

We let [4 = P^ # P^ #... # P^ {h times) be the non-orientable 2-mani- fold of genus h. Recall the elementary fact that the cobordism group 9^2 ÄZ2 and is generated by P^, Thus L4 is cobordant to P^^ for h odd and bounds for h even. Also recall that adding an orientable handle to a non-orientable surface gives the same result as addition of a non- orientable handle (i. e. connected sum with the Klein bottle (/2 = P^ # P^). If K^ cz M^ then clearly an orientable handle can be added to K^ side M^. By these remarks we have that

( 3 . 1 ) U,œM^ => U,^2^M\

Thus questions of embeddabihty break into the two cases h even and h odd, and reduce to the determination, in each of these cases, of the mum h for which (7;, can be embedded in a given 3-manifold.

The map (p: H^{M^\Z2)^^2^^2 c>f Section2 is well-adapted to the parity question since if К^аМ^, then clearly (p(a^^) = genusK (mod 2).