# Principal Bundle

## Cobordisms

Posted in Geometry, Math, Topology by principalbundle on February 22, 2009

A cobordism between two oriented $m$ manifolds $M$ and $N$ is an oriented $(m+1)$ manifold $W$ such that its boundary is $\partial W = \bar{M} \cup N$, where $\bar{M}$ indicates $M$ with reversed orientation.

Why do we have a reversed orientation?  For the same reason we have $\partial (X \times [0,1]) = \bar{X} \times 0 \cup X \times 1$.  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, $\int_{[a,b]} df = f(b)-f(a)$.

We can make this into a stronger relation called an h-cobordism.  An h-cobordism is a cobordism as defined above, where $W$ is homotopically $M \times [0,1]$.  From this we can state,

The h-Cobordism Theorem

Let $M^n$ and $N^m$ be compact, simply connected, cobordant, oriented m-manifolds.  If $m>5$, then there is a diffeomorphism $W = M \times[0,1]$, implying $M$ and $N$ 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.