ON NORMAL EULER NUMBERS OF EMBEDDING SURFACES INTO 4-MANIFOLDS

摘要

Let N be a closed,orientable 4-manifold satisfying H₁（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 H₂（M,（?））,where （?） denotes the twisted integer coefficients determined byw₁（v）=w₁（M）.We study the possible values of e（v）[M],and prove H₁（N-M）=Z₂ or 0.Underthe condition of H₁（N-M,Z）=Z₂,we conclude that e（v）[M]can only take the followingvalues:2σ（N）-2（n+β₂）,2σ（N）-2（n+β₂-2）,2σ（N）-2（n+β₂-4）,…,2σ（N）+2（n+β₂）,where σ（N） is the usual index of N,n the nonorientable genus of M and β₂ 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=S4.this is just the well-known Whitney conjectureproved by W.S.Massey in 1969.