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.     

A SIMPLIFIED APPROACH TO EMBEDDING PROBLEMS IN NORMAL BORDISM FRAMEWORK

The purpose of this paper is to simplify the computations of the nor-mal bordism groups Ω_i(W_f,M×P~∞;(?))and Ω_i(C_f,(?)w;θ_f)which Salomonsenand Dax introduced respectively to study the existence and isotopy classificationof differential embeddings of manifolds in manifolds in the metastable range.Asimpler space pair(K_f,M×P~∞)is constructed to replace(W_f,M×P~∞).It isshown that(K_f,M×P~∞)is homotopy equivalent to(W_f,M×P~∞)and homotopy(n-1)-equivalent to(C_f,(?)W).To demonstrate the efficacy of this simplification,the isotopy groups [M~n(?)RP~(n+k)],if n(?)2k-4 and M~n is a closed(n-k+2)-connected manifold,and[M~n(?)L(p;q_1…,q_m)],if 3n(?)4m-2,M~n is a closed(2n-2m+1)-connected manifold and L is a (2m+1)-dimensional lens space,arespecifically computed.