I am trying to understand the proof of lemma 7.2 in the paper Clifford modules by Atiyah, Bott and Shapiro (Topology, Vol. 3, Supp. 1, 3-38). However I encountered a confusing statement:
Right at the beginning they state ($E$ and $F$ are vector bundles on $X$):
Consider the fibre bundle $Mon(E,F)$ on $X$ whose fibre at $x\in X$ is the space of all monomorphisms $E_x \rightarrow F_x$. This fibre is homeomorphic to $GL(n)/GL(n-m)$ where $n = dim(F), m = dim(E)$, and so it is $(n-m-1)$-connected.
I think that the claim in sentence 2 is wrong and the word homeomorphic should be replaced with homotopic? By dimensional counting I would find that $GL(n)/GL(n-m)$ would have dimension $n^2 - (n-m)^2 = m (2n - m)$, while the space of monomorphisms has dimension $nm$.
The conclusion is not changed and this might just be a typo. Is this a correct observation? If not, what did I get wrong?