## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 45, Number 3 (2001), 823-832.

### 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.

#### Article information

**Source**

Illinois J. Math., Volume 45, Number 3 (2001), 823-832.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138153

**Digital Object Identifier**

doi:10.1215/ijm/1258138153

**Mathematical Reviews number (MathSciNet)**

MR1879237

**Zentralblatt MATH identifier**

0996.57013

**Subjects**

Primary: 57R45: Singularities of differentiable mappings

Secondary: 57Q37: Isotopy 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}

#### Citation

Satoh, Shin. Double decker sets of generic surfaces in $3$-space as homology classes. Illinois J. Math. 45 (2001), no. 3, 823--832. doi:10.1215/ijm/1258138153. https://projecteuclid.org/euclid.ijm/1258138153