# Looking for a reference on the Euler characteristic of the manifold of fixed rank matrices

Let $$\mathcal{M}_r$$ be the set of $$n \times m$$ matrices over $$\mathbb{R}$$ or $$\mathbb{C}$$ of rank $$r$$. What is the Euler characteristic of $$\mathcal{M}_r$$? Can someone point me towards a reference for this calculation?

• Over $\mathbb C$, this should be zero unless $r=0$ because the circle group acts freely on this manifold via multiplication by unit complex numbers. Maybe it's zero over $\mathbb R$ as long as $r\geq 2$ as well. Nov 7 '20 at 2:07
• @WillSawin running the same argument over $\mathbb R$ shows that for $r> 0$, the Euler characteristic is even. Nov 7 '20 at 2:17

$$\mathcal M_r$$ can be described as a fiber bundle over the product $$Gr(r,n) \times Gr(r,m)$$ of the Grassmanian of $$r$$-dimensional subspaces in an $$n$$-dimensional vector space with the Grassmanian of $$r$$-dimensional subspaces in an $$m$$-dimensional vector space, where the fibers are all isomorphic to $$GL_r$$. (These are vector spaces over the field $$\mathbb C$$ or $$\mathbb R$$ as appropriate.) So its Euler characteristic is $$\chi( Gr(r,n) ) \chi( Gr(r,m)) \chi(GL_r).$$
Over the complex numbers $$\chi ( GL_r(\mathbb C))=0$$ for $$r>0$$, making the product zero, and for $$r=0$$ the space $$\mathcal M_r$$ is a point, with Euler characteristic $$1$$.
Over the real numbers, $$\chi(GL_r(\mathbb R))$$ may be calculated by observing that $$GL_r (\mathbb R)$$ maps to $$\mathbb R^r - \{0\}$$ by a fibration whose fibers are all $$\mathbb R^{r-1}$$-bundles over $$GL_{r-1}(\mathbb R)$$, so $$\chi(GL_r(\mathbb R))=\chi(GL_{r-1}(\mathbb R)) \chi ( \mathbb R^r - \{0 \} )$$ which is $$0$$ for $$r \geq 2$$ since $$\mathbb R^2 - \{0\}$$ has Euler characteristic zero.
So the product vanishes for $$r \geq 2$$ and is $$1$$ for $$r=0$$.
Finally, for $$r=1$$, $$\chi(GL_1(\mathbb R))=2$$, and $$Gr(1,n) = \mathbb R \mathbb P^{n-1}$$ which has Euler characteristic $$\frac{ 1 + (-1)^{n-1} }{2}$$, so $$\chi (\mathcal M_1) = \frac{ (1+ (-1)^{n-1} ) (1+ (-1)^{m-1} ) }{ 2}$$ and is always even, as Arun Debray noted.