# Proof of the Euler characteristic of the real Grassmannian $\mathbf{G}(k, n)$

I'm interested in proving the following statement, Let $$\textbf{G}$$(k,n) the real grassmannian and $$\chi_{n,k} := \chi(\textbf{G}(k,n))$$, where $$\chi$$ is the Euler characteristic, then $$\chi_{k,n} = \chi_{k-1,n-1} + (-1)^k\chi_{k,n-1}$$ with initial conditions, $$\chi_{0,n} = \chi_{n,n} = 1$$. My idea for the proof considers the following things:

1. Let the Schubert symbol $$\sigma = (\sigma_1, \ldots ,\sigma_k)$$ for some $$k$$, then

$$\chi_{k,n} = \sum_\sigma (-1)^{\dim e_\sigma}$$

where $$e_\sigma$$ is the Schubert cell associated with $$\sigma$$.

1. The dimension of $$e_\sigma$$ is given by,

$$\dim e_\sigma = \sigma_1 - 1 + \sigma_2 - 2 + \ldots + \sigma_k - k$$

I then proceed as follows,

$$\chi_{k,n} = \sum_{\substack{\sigma \\ \sigma_1 = 1}} (-1)^{\dim e_\sigma} + \sum_{\substack{\sigma \\ \sigma_1 \neq 1}} (-1)^{\dim e_\sigma}$$

Then, if $$\sigma_1 = 1$$,

$$\dim e_\sigma = 1-1 + \sigma_2 - 2 + \ldots + \sigma_k - k$$

so,

$$\chi_{k,n} = \chi_{k-1,n-1} + \sum_{\substack{\sigma \\ \sigma_1 \neq 1}} (-1)^{\dim e_\sigma}$$

This is where I'm stuck, because I don't know if I've done anything wrong so far. So I also want to stress that my intention is to give an ‘elementary’ proof, because I don't want to use homology tools like the Mayer-Vietoris theorem. Any help or suggestions would be greatly appreciated.

If $$\sigma_1 \ne 1$$, then $$\sigma' = (\sigma_1 - 1, \dots, \sigma_k - 1)$$ exactly runs over the symbols for the Schubert cells of $$\mathrm{G}(k, n - 1)$$ but $$\dim e_{\sigma'} = \dim e_\sigma - k$$.