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Can you tell me, where I can find a proof of the following fact: The bordism functor is a generalized cohomology theory, i.e. we can find suitable connecting homomorphisms to obtain long exact cohomology sequences.

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    The connecting homomorphism for the pairs of spaces is one of the easier parts of the proof that it's a (co)homology theory. The proof that bordism is a homology theory (I prefer to say *homology functor* as "theory" IMO is inaccurate and pretentious) is mostly fairly easy -- the part that requires the most care is the proof that it satisfies the excision axiom.2012-07-30

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See Bordism and Cobordism - Atiyah