## Cobordisms

A **cobordism **between two oriented manifolds and is an oriented manifold such that its boundary is , where indicates with reversed orientation.

Why do we have a reversed orientation? For the same reason we have . Which is a consequence of how the orientation of a manifold induces an orientation on the boundary. This is the same as the source of the minus sign in the fundamental theorem of calculus, .

We can make this into a stronger relation called an **h-cobordism**. An h-cobordism is a cobordism as defined above, where is homotopically . From this we can state,

**The h-Cobordism Theorem**

Let and be compact, simply connected, cobordant, oriented m-manifolds. If , then there is a diffeomorphism , implying and are diffeomorphic.

The above statement and proof won Smale a fields medal in the 1960s.

What about in fewer dimensions? In 2-dimensions, this is equivalent to the Poincare conjecture, which has been proven to be true. However in 4-dimensions the theorem is false.

Does this theorem have anything to do with physics? Can string interactions be thought of as cobordisms? From the picture we draw of string interactions, I would not think it was unreasonable if someone told me that interactions could be viewed as cobordisms where the in/out state manifolds aren’t diffeomorphic, but it is not obvious to me if this is actually the case, or if one can do anything useful with it.

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