TL;DR: The state space of an $n$-level quantum system is $\mathbb{CP}^{n-1}$. In particular, the state space of a qubit is $\mathbb{CP}^1$, not $S^3$. Time evolution of a closed quantum system is described by unitary operations. Consequently, if $g:\mathbb{CP}^{n-1}\to\mathbb{CP}^{n-1}$ is an operation that can be realized as the time evolution of a closed quantum system, then $\deg f=1$. In fact, the required homotopy can easily be constructed by interpreting the time parameter as the homotopy parameter and exploiting the physically obvious boundary condition that the zero-duration evolution from $t_0$ to $t_0$ is the identity map.
Quantum state space
Quantum mechanics can be summarized in three to four postulates that identify mathematical objects used to model the states, dynamics and observables of quantum systems. The first postulate describes the state space of an $n$-level quantum system as the complex projective Hilbert space $\mathbb{CP}^{n-1}$.
It is instructive to consider the number of real degrees of freedom in $|c\rangle=c_0|0\rangle + c_1|1\rangle$. The real and imaginary parts of $c_0$ and $c_1$ provide four. Accounting for normalization $|c_0|^2+|c_1|^2=1$ brings the count down to three. At this stage it seems like the qubit's state space might look like $S^3$. It does not. We also need to recognize that multiplying $c_0$ and $c_1$ by any complex number $e^{i\theta}$ with unit magnitude leaves all predictions of the theory unchanged. This is what people mean when they say that the "global phase is unobservable". Indeed, in the density matrix formulation of quantum mechanics the global phase does not make any appearance at all suggesting that it is a feature of the language used to describe phenomena and not a feature of the phenomena themselves. In any case, eliminating the global phase factor, e.g. by choosing $c_0$ to be a non-negative real number, brings the total down to two.
This two-stage procedure gives rise to the equivalence relation
$$
(c_0,c_1)\sim(\lambda c_0,\lambda c_1)\tag1
$$
for any non-zero $\lambda\in\mathbb{C}$. Thus, the numbers $c_0$ and $c_1$, called amplitudes in quantum physics, are in fact just normalized homogeneous coordinates in the complex projective space $\mathbb{CP}^1$. The magnitude of $\lambda$ gives us normalization and the phase angle of $\lambda$ gives us global phase invariance.
The upshot is that the state space of a qubit is $\mathbb{CP}^1\cong S^2$, not $S^3$.
Homotopy between identity and quantum evolution
Another postulate of quantum mechanics says that a closed$^1$ quantum system evolves according to the Schrödinger equation
$$
i\hbar\frac{d}{dt}|\phi(t)\rangle=H(t)|\phi(t)\rangle\tag2
$$
where $H(t)$ is a Hermitian operator corresponding to the system's designated observable called the Hamiltonian. The solution to $(2)$ defines the time evolution operator $U(t_1, t_0)$ which sends a state at time $t_0$ to a state at time $t_1$
$$
|\psi(t_1)\rangle=U(t_1,t_0)|\psi(t_0)\rangle.\tag3
$$
It is physically obvious that we must have
$$
U(t,t)=I\tag4
$$
for every $t\in\mathbb{R}$. Moreover, it can be shown that
$$
U(t_1,t_0)=\mathcal{T}\exp\left(-\frac{i}{\hbar}\int_{t_0}^{t_1}H(t)dt\right)\tag5
$$
where $\mathcal{T}\exp$ is the time-ordered operator exponential.
The boundary condition $(4)$ suggests a way of constructing a homotopy between the identity and any map that describes a physically realizable time evolution. Indeed, consider a map $g:\mathbb{CP}^{n-1}\to\mathbb{CP}^{n-1}$. By the above postulate, if $g$ describes a physically realizable time evolution of an $n$-level quantum system, then there exists a Hamiltonian $H(t)$ and $t_0,t_1\in\mathbb{R}$ such that the time evolution operator corresponding to the solution of Schrödinger equation satisfies
$$
U(t_1, t_0)=g.\tag6
$$
Define $G:\mathbb{CP}^{n-1}\times[0,1]\to\mathbb{CP}^{n-1}$ as
$$
G(|\psi\rangle, t):=U(t_0+(t_1-t_0)\cdot t, t_0)|\psi\rangle\tag7
$$
for every $|\psi\rangle\in\mathbb{CP}^{n-1}$ and every $t\in[0,1]$. By $(4)$ we have $G(.,0)=I$ and by $(6)$ we have $G(.,1)=g$. Moreover, $(5)$ implies that $G$ is continuous. Therefore, $G$ is a homotopy between the identity and $g$. Finally, since the degree is homotopy invariant, we have
$$
\deg g=\deg I=1.\tag8
$$
Thus, every physically realizable function on $\mathbb{CP}^{n-1}$ has degree $1$.
$^1$ Here, "closed" means that the system is isolated from its environment.
