We give examples of open –manifolds and –orbifolds that exhibit pathological behavior with respect to splitting along surfaces (–suborbifolds) with nonnegative Euler characteristic.
"Some open $3$–manifolds and $3$–orbifolds without locally finite canonical decompositions." Algebr. Geom. Topol. 8 (3) 1795 - 1810, 2008. https://doi.org/10.2140/agt.2008.8.1795