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date: 2021-04-26
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#homotopy #higher-algebra/cobordism 

# Why study cobordism?

[Cobordism](Cobordism.md) is a much coarser equivalence relation than diffeomorphism or homeomorphism of [manifolds](manifolds), and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 - because the word problem for groups cannot be solved - but it is possible to classify manifolds up to cobordism.

It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the [Hirzebruch-Riemann-Roch theorem](Hirzebruch-Riemann-Roch%20theorem), and in the first proofs of the [Atiyah-Singer index theorem](Atiyah-Singer%20index%20theorem).

Every vector bundle theory (real, complex etc.) has an extraordinary cohomology theory called K-theory