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For a closed smooth manifold, we can use the Stiefel-Whitney number of the manifold to decide it is boundary or not.

For a closed topology or PL manifold, how to decide it is a boundary of compact Top/PL manifold?

As far as I know, the Top/PL cobordism ring is a litter complex. So I guess they may not have simple way as in the smooth case to decide which closed Top/PL manifold is boundary. I hope I am wrong.

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  • $\begingroup$ I don't know if it completely answers your question, but you may be interested in Brumfiel, Madsen, and Milgram, "PL characteristic classes and cobordism." $\endgroup$ Oct 20, 2017 at 14:13
  • $\begingroup$ Thank you for your answer. In fact, since I can't find the exactly answer in Madsen's work, then I pose the question here. $\endgroup$ Oct 20, 2017 at 14:36

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