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In [3], after defining notions of LS category in the simplicial context, theauthors show that the geometric simplicial LS category is non-decreasing understrong collapses. However, they do not give examples where it increasesstrictly, but they conjecture that such an example should exist, and thus thatthe geometric simplicial LS category is not strong homotopy invariant. Thepurpose of this note is to provide with such an example. We construct asimplicial complex whose simplicial and geometric simplicial LS categories aredifferent, and using this, we provide an example of a strong collapse thatincreases the geometric simplicial LS category, thus settling the geometricsimplicial LS category not being strong homotopy invariant.

A strong collapse increasing the geometric simplicial Lusternik-Schnirelmann category