Does anybody have a good references for fact that stable homotopy groups forms generalized homology theory (in sense of Eilenberg–Steenrod axioms)?

basic quick question outside my field: Is the kernel of the diract operator on a threedimensional manifold zero always?

No, for instance the threefold product of the periodic (i.e. non-bounding) spin structure on S^1 should have nonvanishing kernel (just tensor up sections in the kernel of the Dirac operator on S^1). However since the Atiyah orientation MSpin -> KO is an iso in this dimension the Â-genus of every three-dimensional spin manifold vanishes, so the kernel and cokernel have the same dimension (more precisely, a perturbation of the Dirac operator has no kernel)

Thank you very much!

Why is it that there's a homotopy chat room but say not a general MO chatroom?

@fpqc There is or was a general MO chatroom, just not very active. This one became active largely because of a group / clique of people in the same research area who all knew each other personally and had reasons to talk regularly and sources of questions of common interest to discuss. I'm not sure that there's a very strong bond between general MO users.