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Abstract

We study the sensibility of an invariant of 2-dimensional CW complexes in the case when it comes as a reduction (through a change of ring) of a modular invariant of 4-dimensional thickenings of such complexes: it is shown that if the Euler characteristic of the 2-complex is greater than or equal to 1, its invariant depends only on homology. To see what is happening when the Euler characteristic is smaller than 1, we use ideas of Kerler and construct, from any tortile category, an invariant of 4-thickenings which, if the category is modular, is a normalization of the Reshetikhin–Turaev invariant, and if the category is symmetric, depends only on the spine and the second Whitney number of the thickening. Then we show that the ${\rm so}(3)$ quantum invariant at a 5th root of unity has its reduction, and this reduction is able to distinguish complexes with the same homology groups when the Euler characteristic is smaller than 1.