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Is there a good reference for the proof that the cobordism group of pseudo-manifolds is isomorphic to the singular homology group?

I was looking for a more geometrical definition of homology and found these notes that give a more geometrical definition of homology and I also found an entry in the Encyclopedia of Mathematics that gave a reference for the proof of the result above. However, I would want to know if there is a more modern and easy to read reference.

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You should be aware of the book "A geometric approach to homology theory" by Buoncristiano, Rourke and Sanderson. However, I am not willing to claim that it is modern (1976) or easy to read.

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For a modern textbook reference, you could try On Thom Spectra, Orientability and Cobordism by Yuli Rudyak. The result you mention is in Chapter VIII.

You may also be interested in the presentation of singular (co)homology as (co)bordism of stratifolds, due to Matthias Kreck. His textbook Differential Algebraic Topology is extremely well-written, and reads like a textbook for an alternative first course on homology theory.

Both of these books can be found online without too much effort.

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