Double decker sets of generic surfaces in $3$-space as homology classes

Abstract

The double decker set $\Gamma$ of a generic map $g:F_0^2\rightarrow M^3$ is the preimage of the singularity of the generic surface $g(F_0)$. If both $F_0$ and $M$ are oriented, then $\Gamma$ is regarded as an oriented 1-cycle in $F_0$, which is shown to be null-homologous if $g(F_0)=0\in H_2(M;{\mathbf Z})$. We also investigate a double decker set of a surface diagram which is a generic surface in ${\mathbf{R}}^3$ with crossing information.