Almost complex structures on 8-manifolds
The main purpose of this paper is to give a criterion for the existence of almost complex structures on 8-dimensional manifolds. Several theorems are proved, we give here the following. Writing \(TH^2(M)\) for the torsion subgroup \(H^2(M,\mathbb{Z})\), the following three cases are distinguished: case \(\text{I}:\rho_2^{-1}(w_2(M))\cap TH^2(M)=\emptyset\). case \(\text{II}_+: \rho_1^{-1} (w_2(M))\cap TH^2(M)\neq \emptyset\) and \(b_2(M)>0\). case \(\text{II}_0: \rho_1^{-1} (w_2(M))\cap TH^2(M)\neq \emptyset\) and \(b_2(M)=0\). Theorem: (a) An oriented 8-dimensional manifold \(M\) admits a stable almost complex structure if and only if \(\beta w_2(M)=0\) and, in case II, \(\chi(M)= 0\bmod 2\). (b) \(M\) admits an almost complex structure if and only if the conditions of (a) hold and (i) \(\chi(M)= \tau(M)\bmod 4\) in cases I and \(\text{II}_+\); (ii) \(8\chi (M)-4p_2(M)+ p^2_1(M)=0\) in case \(\text{II}_0\).
scientific article; zbMATH DE number 1558752