**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 $[g\_i]\in H\_\*(V\_k(\mathbb{R}^N))$.



Homotopy class of maps into Stiefel manifolds
Maybe what you looking for is known under the name *generalized curvatura integra* (for the case $N> k+1$). I will formulate it not for $S^n$ but more generally for a $m$-dimensional framed manifold $M$, i.e. there is an embedding $F \colon M\to \mathbb R^{m+k}$ with trivialized normal bundle $\nu(F)\cong\varepsilon ^k$. This gives a map
$$
c\colon M \to V\_k(\mathbb R^{m+k}),\quad p\mapsto \nu(F)\_p\cong \mathbb R^k \subset T(\mathbb R^{m+k})\_{F(p)}\cong \mathbb R^{m+k}.
$$


**Definition:** The generalized curvatura integra (gci) s defined by
$$
c\_\ast[M] \in H\_m(V\_{m+k,k}) = 
\begin{cases}
\mathbb Z &m \equiv 0 \mod 2\\
\mathbb Z\_2 & m\equiv 1 \mod 2.
\end{cases}
$$


Kervaire computed the gci in [Relative characteristic classes](https://www.jstor.org/stable/2372561?origin=crossref&seq=1#metadata_info_tab_contents) and [Courbure integrale generalisee et homotopie](https://link.springer.com/article/10.1007%2FBF01342961) as follows:
$$
c\_\ast[M] = H(M,F) +
\begin{cases}
\chi(M)/2 \in \mathbb Z, \,m\equiv 0\mod 2\\
\chi\_{1/2}(M) \in \mathbb Z,\, m\equiv 1\mod 2\\
\end{cases}
$$


where 


* $\chi(M)$ is the Euler characteristic
* $\chi\_{1/2}(M)$ is the *Kervaire semicharacteristic* and is defined as
$$
\chi\_{1/2}(M)= \sum\_{j=0}^{(m-1)/2} \dim\_{\mathbb Z\_2} H\_j(M,\mathbb Z\_2) \mod 2 \in \mathbb Z\_2
$$
* $H(M,F)=0$ if $m \equiv 0 \mod 2$ and in case of $m \equiv 1\mod 2$ the number $H(M,F)$ is
defined as follows: Since $M$ is framed we have by the Pontryagin-Thom construction an induced
map $\tilde F \colon S^{k+m} \to S^k$. $H(M,F)$ is the determined by the Steenrod square in the mapping cone $X = S^k \cup\_{\tilde F} D^{m+k+1}$, i.e. if $x \in H^k(X;\mathbb Z\_2)$ and $y \in H^{k+m+1}(X;\mathbb Z\_2)$ are generators then 
$$
Sq^{m+1}(x) = H(M,F)\cdot y.
$$


