An extension of Whitney's congruence
type: Item
instance of: scholarly article
title: An extension of Whitney's congruence
zbMATH Open document ID: 0855.57018
author: Yuichi Yamada
published in: Osaka Journal of Mathematics
publication date: 1997-02-03T00:00:00Z
review text: 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 \(H_2 (M; Z_2)\to Z_4\).
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 755445
zbMATH Keywords: Whitney's congruence
zbMATH Keywords: PL 4-manifold
zbMATH Keywords: locally flat surface
zbMATH Keywords: intersection form
zbMATH Keywords: flat embedding
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 755445