Arithmetic and Complex Bordism

Bordism theories are particular sorts of homology theories, built in accordance to the rule singular homology : simplices :: bordism : manifolds. Fixing a structure group G, the geometric chain complex GC∗(X ;G) of a space X is given by GCn(X ;G) =N f : M →X M a connected manifold with G-structure . The boundary maps in this complex are induced by the restriction of f to the boundary of M . Hence, the n-cycles of this complex are given by the maps off closed n-manifolds, and the n-boundaries are given by those maps which extend to a map from an (n+ 1)-manifold. This complex describes the value of the homology theory Ω∗ (X ) for any space X , but to begin to grapple with it we should first consider the case X = pt. Since equipping a manifold with a map to a point is no extra data at all, the bordism complex of a point is just the complex generated by the manifolds themselves, with boundary maps induced by taking the boundary submanifold. It’s instructive to draw at least one picture: in the case where there’s no structure group (i.e., when G is the orthogonal group), we can consider the following 2-manifold with boundary: