# Homotopy class of maps into Stiefel manifolds

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

• Could you give us a little more detail on the gap between what you want and the homotopy group of a Stiefel manifold? – Ryan Budney Dec 30 '19 at 19:42
• Maps from $S^n$ into $O(k)$ correspond to real vector bundles over $S^{n+1}$ of rank k. – Connor Malin Dec 30 '19 at 20:34
• @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 $[M_i] \in H^*(V_k(R^n))$. – Warlock of Firetop Mountain Dec 31 '19 at 9:00

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: $$c_\ast[M] = H(M,F) + \begin{cases} \chi(M)/2 \in \mathbb Z, \,m\equiv 0\mod 2\\ \chi_{1/2}(M) \in \mathbb Z,\, m\equiv 1\mod 2\\ \end{cases}$$
• $$\chi(M)$$ 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 $$Sq^{m+1}(x) = H(M,F)\cdot y.$$