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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 $[g_i]\in H_*(V_k(\mathbb{R}^N))$.

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    $\begingroup$ Could you give us a little more detail on the gap between what you want and the homotopy group of a Stiefel manifold? $\endgroup$ – Ryan Budney Dec 30 '19 at 19:42
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    $\begingroup$ Maps from $S^n$ into $O(k)$ correspond to real vector bundles over $S^{n+1}$ of rank k. $\endgroup$ – Connor Malin Dec 30 '19 at 20:34
  • $\begingroup$ @RyanBudney , I am looking for a map (maybe similarly to the degree) that given a function defined over the sphere returns an element in the homotopy group corresponding to the homotopy class of the map. Since the classification of maps into Steifel manifolds is complicated, I expect to have more invariants for example of the form $\langle f^*[M_i], [\mathbb{S}^n]\rangle$ where $[M_i] \in H^*(V_k(R^n))$. $\endgroup$ – Warlock of Firetop Mountain Dec 31 '19 at 9:00
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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 and Courbure integrale generalisee et homotopie 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 $$ Sq^{m+1}(x) = H(M,F)\cdot y. $$
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