## Günter Rote:

# Piecewise linear Morse theory

*In: Oberwolfach Reports, 3, European Mathematical Society -
Publishing House, 2006, pp. 696–698. *doi:0.4171/OWR/2006/12
### Abstract

Classical Morse Theory considers the topological changes of the level
sets
`M`_{h}={`x` in `M` | `f`(`x`)=`h`}
of a smooth function `f` defined on a
manifold `M` as the height `h` varies. At critical
points, where
the gradient of `f` vanishes, the topology changes. These changes
can be
classified locally, and they can be related to global topological
properties
of `M`. Between critical values, the level sets vary smoothly. We
prove
that the same statement is true for *piecewise linear*
functions in up
to three variables: between critical values, all level sets are
isotopic.
Last update: June 12, 2007.