TITLE: Trivial cobordism group in dimensions 1, 3, 7 related to H-space structures on the spheres in these dimensions?
QUESTION [7 upvotes]: Is there a connection between the existence of H-space structures on $S^1$, $S^3$ and $S^7$ and the fact that every (closed) 1-manifold, 3-manifold and oriented 7-manifold is a boundary, or is this a pure numerical coincidence? 
[It's possible that my question amounts to groundless mysticism, but I had to wonder.]
On a related note: is it known which (oriented or not) cobordism groups are zero?

REPLY [3 votes]: How about $n\ne 0,2,6$ instead of $n=1,3,7$? The Adams theorem on the vanishing of the stable Hopf invariant, which implies the non-existence of an H-structure on $S^n$ for $n\ne 1,3,7$, also implies the theorem of R. L. W. Brown and Liulevicius (and Mahowald) that if $M^n$ immerses in $\Bbb R^{n+1}$, then $M^n$ is a boundary or is cobordant to $\Bbb RP^0$, $\Bbb RP^2$ or $\Bbb RP^6$ (and these three do immerse in $\Bbb R^{n+1}$).