# Cobordism theory of some weird space

Let $$G=SU(3)$$ and $$N=SO(3)$$, then $$G/N= SU(3)/SO(3)$$ = a 5-dimensional Wu manifold $$W$$.

The $$W$$ is a homogeneous space (also a quotient space), but not a group.

Previously, I am aware of the bordism group of $$BG$$ where $$BG$$ is the classifying space of a group $$G$$. The $$d$$-th bordism group is $$\Omega_d^H (BG)$$.

Now let us consider the quotient of their classifying spaces $$BSU(3)/BSO(3)$$ (either the point set quotient or the mapping cone).

I am not aware of the bordism group of $$\Omega_d^H(BSU(3)/BSO(3))$$ of this space $$BSU(3)/BSO(3)$$. Are there previous knowledge of computing the bordism group of this type?

Here $$H$$ can be some structures equipped for the cobordant of manifolds.

• As far as I know, nobody has computed these bordism groups. I would guess the standard spectral-sequence approaches that you already know probably work, but I'm not certain. – Arun Debray May 24 at 17:30
• There's also the long exact sequence for a cofibration...does that not tell you anything useful? – John Greenwood May 24 at 21:36