Gao Hongzhu
Journal of Systems Science and Complexity. 1990, 3(2): 166-171.
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) of v is an element of H_2(M,\mathcal{X}), where \mathcal{X} denotes the twisted integer coefficients determined by w_1(v)=w_1(M). We study the possible values of e(v)[M], and prove H_1(N-M)=Z_2 or 0. Under the condition of H_1(N-M,Z)=Z_2, we conclude that e(v)[M]can only take the following values: 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 embedding for N=homological sphere. In the case of N=S~4. this is just the well-known Whitney conjecture proved by W.S.Massey in 1969.
