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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.

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  • $\begingroup$ 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. $\endgroup$
    – Arun Debray
    Commented May 24, 2020 at 17:30
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    $\begingroup$ There's also the long exact sequence for a cofibration...does that not tell you anything useful? $\endgroup$
    – John Greenwood
    Commented May 24, 2020 at 21:36

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