Here is the actual paper: http://arxiv.org/pdf/1303.2354v4.pdf http://arxiv.org/pdf/1303.2354v4.pdf. It looks like this is their family of counter examples: A

Here is the actual paper: http://arxiv.org/pdf/1303.2354v4.pdf http://arxiv.org/pdf/1303.2354v4.pdf. It looks like this is their family of counter examples: A specific non-orientable five-dimensional manifold M5 with Sq1 ∆(M) ̸= 0 is constructed in [25]. Hence M5 is non-triangulable. To get non-orientable examples of non-triangulable manifolds in dimensions n > 5 we can take the product of M5 with the torus Tn−5. To get an orientable example in dimension 6 we can consider the non-orientable S1-bundle over M given by M ̃ ×Z/2 S1, where M ̃ → M is the oriented double cover; the total space of this bundle is orientable. To get orientable examples in dimensions n > 6, we can then take products with Tn−6.