# Delooping the quotient space $SU/SU(n)$

Let $$SU$$ denote the infinite unitary group. Does the quotient space $$SU/SU(n)$$ admit a delooping $$X$$? One could also ask that this space $$X$$ sit in a fiber sequence $$BSU(n)\to BSU\to X$$, but this is not strictly part of the question. Note that $$SU/SU(n)$$ is not a topological group, because $$SU(n)$$ is not normal in $$SU$$ --- but this doesn't prohibit $$SU/SU(n)$$ from admitting a delooping. Perhaps a geometric construction of a H-space structure can be given by viewing the space as a Stiefel manifold. Note that $$SU$$ is an infinite loop space by Bott periodicity.

I'll work with mod $$2$$ cohomology. Note that $$H^*(BSU(2))$$ is polynomial on $$c_2$$ (in degree $$4$$) and $$H^*(BSU)$$ is polynomial on $$c_k$$ for $$k\geq 2$$. Here $$c_k$$ has degree $$2k$$ and so $$H^6(BSU)=\{0,c_3\}$$. If $$X$$ exists then it seems we should have $$H^*(X)$$ polynomial on generators in degrees $$6,8,10,\dotsc$$. In particular, $$H^6(X)$$ should be generated by $$c_3$$ and $$H^{10}(X)$$ should be generated by a single element that is $$c_5$$ modulo decomposables. However, $$H^*(X)$$ should also be closed under the action of the Steenrod algebra so it should contain $$\text{Sq}^4(c_3)$$, which is $$c_2c_3$$ by a calculation with symmetric polynomials. This is inconsistent, so $$X$$ cannot exist. I would guess that this line of argument can be improved to show that $$SU/SU(2)$$ does not deloop, but I have not tried to work out the details.
• Thanks. I guess the equation $\mathrm{Sq}^4(c_m) = c_2 c_m$ shows more generally that there's no delooping of $SU/SU(n)$ with a map from $BSU$ whose homotopy fiber is $BSU(n)$.