Principal Bundle


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.