GL(2,ℝ) is not simply connected

Idea. A space and a deformation retract of it have the same fundamental group.

Step 1: path connected. Proved in Problem 8 (together with Problem 6 for $SO(2)$).

Step 2: $SO(2)$ is a circle. The map $\theta\mapsto R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$ induces a homeomorphism $\mathbb{R}/2\pi\mathbb{Z}\to SO(2)$ (continuous bijection from a compact space to a Hausdorff space, Problem 6). So $SO(2)\cong S^1$ and $\pi_1(SO(2),I)\cong\mathbb{Z}$, generated by the loop $\ell(\theta)=R_{2\pi\theta}$, $\theta\in[0,1]$.

Step 3: deformation retraction. By Problem 25, $SO(2)$ is a strong deformation retract of $GL_2^+(\mathbb{R})$: there is a continuous $H:GL_2^+\times[0,1]\to GL_2^+$ with $H(\cdot,0)=\mathrm{id}$, $H(\cdot,1)=r$ where $r:GL_2^+\to SO(2)$ is a retraction ($r|_{SO(2)}=\mathrm{id}$). Explicitly $H(A,t)=U(A)((1-t)P(A)+tI)$ with $A=U(A)P(A)$ the polar decomposition. Hence the inclusion $i:SO(2)\to GL_2^+$ induces an isomorphism
$$i_*:\pi_1(SO(2),I)\xrightarrow{\ \cong\ }\pi_1(GL_2^+,I).$$

Step 4: conclusion. $\pi_1(GL_2^+(\mathbb{R}),I)\cong\mathbb{Z}\ne0$. The loop $\ell(\theta)=R_{2\pi\theta}$ is a generator and is not null-homotopic in $GL_2^+$.

A direct invariant. The retraction gives an explicit "winding number": to a loop $\gamma$ in $GL_2^+$ associate the loop $r\circ\gamma$ in $SO(2)\cong S^1$; the winding number is the degree. Equivalently, for $\gamma(\theta)=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, it is the winding number around $0$ of the complex number $(a+d)+i(c-b)$, which never vanishes when $\det\gamma>0$ (indeed $|a+d|^2+|c-b|^2=(a^2+b^2+c^2+d^2)+2\det\gamma>0$).

Remark. For $n\ge3$, $\pi_1(SO(n))=\mathbb{Z}/2$ (the "plate trick"), so $\pi_1(GL_n^+)=\mathbb{Z}/2$: for $n=2$ the fundamental group is $\mathbb{Z}$, which is exceptional. The universal cover of $GL_2^+(\mathbb{R})$ is a non-matrix group.