John Baldwin will give the talk on Monday, October 25 at 4:15 in Math 528. This lecture will be open to all.
Abstract:
The oriented cobordism group On consists of equivalence classes
of smooth closed n-manifolds, where X and Y are equivalent iff X "disjoint
union" -Y is the boundary of a smooth (n+1)-manifold. (Here, -Y is simply Y
with the reversed orientation). The group operation is disjoint union.
Furthermore, we can form the oriented cobordism ring, O*, which is the
direct sum of the On, for n from 1 to infinity. The ring multiplication
here is cartesian product. I will demonstrate a correspondence between On
and pi_m (T), the mth homotopy group of T, for m large, where T is the "Thom
space" associated to the canonical bundle over the oriented grassmannian.
This is a mouthful, but all of these spaces will be defined, and the talk
will be accessible to undergraduates.
If time permits, I will sketch a way to calculate from this the tensor
product of O* with the rational numbers. It turns out that the ring O*
tensor Q is freely generated by the manifolds CP2, CP4, CP6, ...
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