ON NORMAL EULER NUMBERS OF EMBEDDING SURFACES INTO 4-MANIFOLDS

Let N be a closed,orientable 4-manifold satisfying H_1(N,Z)=0,and M be a closed,connected,nonorientable surface embedded in N with normal bundle v.The Euler class e(v)ofv is an element of H_2(M,(?)),where (?) denotes the twisted integer coefficients determined byw_1(v)=w_1(M).We study the possible values of e(v)M],and prove H_1(N-M)=Z_2 or 0.Underthe condition of H_1(N-M,Z)=Z_2,we conclude that e(v)M]can only take the followingvalues:2σ(N)-2(n+β_2),2σ(N)-2(n+β_2-2),2σ(N)-2(n+β_2-4),…,2σ(N)+2(n+β_2),where σ(N) is the usual index of N,n the nonorientable genus of M and β_2 the 2nd real Bettinumber.Finally,we show that these values can be actually attained by appropriate embeddingfor N=homological sphere.In the case of N=S~4.this is just the well-known Whitney conjectureproved by W.S.Massey in 1969.

ON NORMAL EULER NUMBERS OF EMBEDDING SURFACES INTO 4-MANIFOLDS