23 - Summary on complex cobordism

Summary

The next piece is a summary on complex cobordism, written by me especially for the present work.

The subject of complex cobordism has two aspects: a geometrical side, on which it links up with the theory of complex manifolds, and a hornotopy-theoretic side, on which it links up with generalised homology and cohomology theories and the study of spectra. I begin by sketching this. (Afterwards I will present the calculation of π(MU), and finish by sketching some topics from the further development of the subject.)

Let M^m₁ and M^m₂ be two smooth manifolds, of dimension m, compact, without boundary and both of the same sort: that is, both non-oriented, or both oriented, or both with whatever extra structure is to be considered. Then we say that M^m₁ and M^m₂ are cobordant if there is a smooth manifold W^m+1 of dimension m + 1, compact, with boundary, and of the same sort, such that the boundary of W^m+1 is the disjoint union of M^m₁ and M^m₂.

Here the notion of ‘boundary’ is taken in the sense appropriate to manifolds of the sort considered, so that we include a condition on any extra structure we may have. For example, if we are working with oriented manifolds, then we ask that the boundary of the orientation class on W^m+1 should be plus the orientation class on M^m₂, minus the orientation class on M^m₁. Similarly for other forms of extra structure. Cobordism is an equivalence relation, and divides m-manifolds of the sort considered into equivalence classes.