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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 that doesfor this calculation?

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 that does this calculation?

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?

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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 that does this calculation?