Let us define the set of maps $$M_d:= \left\{R: (I^d,\partial I^d)\to (V_p(\mathbb{C}^q),x_0)\mid \overline{R(k)} = R(-k)\right\} \subseteq \left\{ R: (I^d,\partial I^d)\to (V_p(\mathbb{C}^q),x_0) \right\} = \Omega^d(V_p(\mathbb{C}^q))$$ where $I = [-1,1]$ and $V_p(\mathbb{C}^q)$ is the complex Stiefel manifold whose elements we will interpret as complex $q\times p$-matrices whose columns are mutually orthonormal.

We equip the sets $M_d$ and $\Omega^d(V_p(\mathbb{C}^q))$ with the Compact-open topology, such that they become topological spaces and would like to calculate the homotopy groups $\pi_n(M_d,c_{x_0})$, where $c_{x_0}$ is the constant map into $x_0$.

My problem is that I have no Ansatz in computing $\pi_n(M_d,c_{x_0})$. What I do know are the homotopy groups $\pi_n(\Omega^d(V_p(\mathbb{C}^q)),c_{x_0})\cong \pi_{n+d}(V_p(\mathbb{C}^q),x_0)$, which is a standard result in homotopy theory. But $M_d$ is a subspace in $\Omega^d(V_p(\mathbb{C}^q))$, which does not have to share the same homotopy groups. The elements of $M_d$ satisfy a certain $\mathbb{Z}_2$-equivariance condition and the theory about $G$-equivariant homotopy seems to be very involved, although I would certainly dive into it, when I knew that there are tools with which one could calculate the homotopy groups of $M_d$.

I came across a PhD-thesis in physics, "Homotopy Theory of Topological Insulators" by Ricardo Kennedy, who claims something like $\pi_n(M_d,c_{x_0})\cong\pi_{n+1}(M_{d-1},\Omega^{d-1}(V_p(\mathbb{C}^q),c_{x_0})$ in Lemma 3.7. Is this correct and is there further literature about this?

The space $M_d$ is basically an iterated loop space with an additional $\mathbb{Z}_2$-equivariance condition, which seems to make life hard.

Thank you in advance!