**Motivation**

Hopf theorem, asserts that $C^0$-maps $f:M^n\to \mathbb{S}^n$ from an orientable, closed n-manifold into an n-sphere are classified up to homotopy by their degree $deg(f)$. The theorem not only says that $[\mathbb{S}^n, \mathbb{S}^n] \simeq \mathbb{Z} $ but also gives us a way to compute the complexity of the map, namely the degree. I am looking for a similar invariant of maps into Stiefel manifolds and orthogonal groups (they should be related).

1)Consider a map $f:\mathbb{S}^n \to V_k(\mathbb{R}^N)$ where $V_k(\mathbb{R})^n$ is the Stiefel manifold of $k$-orthogonal frames of $\mathbb{R}^N$. Is there an invariant $\mathcal{I}(f)$ that similarly to the degree, provides us with a correspondence with the homotopy classes of maps $[\mathbb{S}^n , V_k(\mathbb{R}^N)]$?

2) What can we say with the orthogonal group $O(k)$ in place of $V_k(\mathbb{R}^N)$? (This should be related)

**What we are looking for**

Of course, if $N>k+1$ then $V_k(\mathbb{R}^N)$ is simple and $[\mathbb{S}^n , V_k(\mathbb{R}^N)]\sim \pi_n V_k(\mathbb{R}^N)\simeq \mathbb{Z} \text{ or } \mathbb{Z}/2$
but this is not enough, we need to pick a generator and once we have done this *how do we associate to a function a multiple of the generator?*.

As the degree of $f:M^n\to \mathbb{S}^n$ can be defined homologically ($f_*[M^n]= \deg(f)[\mathbb{S}^n]$), I expect that for our map $f:\mathbb{S}^n\to V_k(\mathbb{R}^N)$ we can use something like a set of integers $\langle f_*[\mathbb{S}^n],[g_i] \rangle\in \mathbb{Z}$ where $[g_i]\in H_*(V_k(\mathbb{R}^N))$.