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Sep 26, 2018 at 18:03 comment added mme @annieheart You can define "bordism groups" for pretty much any kind of manifolds you like. Normally one means smooth manifolds. Certainly you can make sense of it for non-triangulable manifolds, but these are necessarily non-smoothable. So you would want to know about $\Omega_*^{\text{Top}}$; there is some study of this in a book by Madsen and Milgram (classifying spaces for...). You might find this answer useful.
Sep 26, 2018 at 14:55 comment added annie marie cœur Does bordism/cobordism classify the cobordant of ONLY triangulated manifolds, or does it classify the cobordant of non-triangulated manifolds as well?
May 12, 2017 at 0:47 history edited mme CC BY-SA 3.0
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Mar 11, 2017 at 1:56 vote accept Joseph O'Rourke
Mar 11, 2017 at 1:40 history answered mme CC BY-SA 3.0