An extension of Whitney's congruence
Recall Whitney's congruence formula for a connected closed locally flat surface \(F\) in \(S^4\): \[ e(S^4, F)+ 2\chi (F)\equiv 0\bmod 4, \] where \(e(S^4, F)\) is the normal Euler number of embedding \(F\subset S^4\) and \(\chi (F)=\) Euler characteristic. The paper gives the right hand side when the ambient space is a closed connected and oriented PL 4-manifold \(M\). So if \(M\) is such a PL 4-manifold with \(H_1 (M; Z)=0\) and \(F\subset M\) a closed, connected and locally flat surface, then it holds \[ e(M, F)+ 2\chi (F)\equiv q([ F])\bmod 4, \] where \(q([ F])\) means the following: Let \(p_2: H_2 (M; Z)\to H_2 (M; Z_2)\) be the mod 2-reduction, then \(q: H_2 (M; Z_2)\to Z_4\) is a quadratic function defined as \(q(\alpha)= \widetilde {\alpha} \circ \widetilde {\alpha} \bmod 4\), where \(\widetilde {\alpha}\) is an element of \(p_2^{-1} (\alpha)\) and \(\circ\) is the intersection form on \(H_2 (M; Z_2)\). Relaxing the condition \(H_1 (M; Z)=0\), it is obtained: If \(M\) is an oriented PL 4-manifold, then the assignment of \(e(M, F)+ 2\chi (F)\bmod 4\) to locally flat embedding \(F\subset M\) induces a \(Z_4\)-quadratic map from \(H_2 (M; Z_2)\to Z_4\).
scientific article; zbMATH DE number 755445